PSPM SP015
PSPM CHAPTER 6: CIRCULAR MOTION
___________________________________________________________________________
Uniform Circular Motion
PSPM 2017/2018 SF016/2 No. 4(a)(ii)
1. Define centripetal acceleration. [1 m]
UPS 2001/2002 SF015 No. 5(a)
2. Does a body moving in a uniform circular motion have acceleration? Explain briefly.
[2 m]
PSPM 2013/2014 SF016/2 No. 4(a)(ii)
3. Why is the direction of centripetal acceleration always perpendicular to the velocity?
[2 m]
PSPM 2016/2017 SF016/2 No. 4(a)
4.
P Q
FIGURE 6.1
Sketch the change in the velocity of an object due to position P and Q which will show
that the object is in a uniform circular motion as shown in FIGURE 6.1. [2 m]
PSPM 2008/2009 SF017/2 No. 11(b)
5. Calculate the centripetal acceleration of an object at the equator due to rotation of earth.
6
The radius of earth is 6.4 × 10 m. [4 m]
Centripetal Force
PSPM 2013/2014 SF016/2 No. 4(a)(i)
6. Define centripetal force. [1 m]
1
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2014/2015 SF016/2 No. 4(a)
7. A body is performing a uniform circular motion.
(a) Explain why the body experiences a force despite moving with constant speed.
[2 m]
(b) Write an equation of the force and state its direction. [2 m]
PSPM 2011/2012 SF016/2 No. 4(a)
8. Name the force that is responsible for the following motion:
(a) A satellite orbiting earth. [1 m]
(b) A ball attached to a string that swirls horizontally. [1 m]
PSPM 2012/2013 SF016/2 No. 4(a)
9. Name the centripetal force that is responsible for the following motions:
(a) A car moves around a curve without skidding. [1 m]
(b) The moon orbiting the earth. [1 m]
PSPM JAN 1999/2000 SF015/2 No. 5 Edited
10. A
rotating wheel
D B
chair
C
FIGURE 6.2
FIGURE 6.2 shows a passenger riding a vertical rotating wheel of 4.0 m diameter
rotating at constant speed.
(a) Write equations for the reaction force acting on the passenger at position A and B.
[2 m]
(b) What is the maximum angular speed so that the passenger is not thrown out?
[2 m]
2
PSPM SP015
PSPM JUN 1999/2000 SF015/2 No. 4
-1
11. An object of 2.5 kg mass is rotated in a vertical circle at a speed of 5.0 m s . Determine
the centripetal force of the object. Also calculate the maximum and minimum tension of
the string between the object and the centre of the circle if the radius of the circle is
2.0 m. [4 m]
UPS 2001/2002 SF015 No. 5(a)
12.
FIGURE 6.3
A body of mass 2.3 kg is hung freely from a piece of string of 162 cm long. The body is
circulated horizontally so that the string makes an angle with the vertical line as
shown in FIGURE 6.3.
(a) Draw all the forces acting on the body. [2 m]
(b) If the tension of the string cannot exceed 50 N, calculate the maximum angle so
that the string does not break and its maximum speed. [6 m]
PSPM 2003/2004 SF017/2 No. 2
13.
r
FIGURE 6.4
A pendulum with length L makes a circular motion in the horizontal plane as shown in
FIGURE 6.4. If the cord of the pendulum makes an angle with the vertical axis, find
the rotating frequency of the pendulum in terms of its radius r, , and acceleration due
to gravity, g. [4 m]
3
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2003/2004 SF017/2 No. 11(b)(i), (ii), (iii)
14.
Z
FIGURE 6.5
FIGURE 6.5 shows a particle at rest on the highest part of a slippery hemisphere of
radius r. When released, the particle slides downwards until it breaks loose from the
hemisphere surface.
(a) Draw all the forces acting on the particle after it is released and when it is still
touching the hemisphere. [2 m]
(b) Based on your drawing in (a), what is the resultant force of these forces. [1 m]
(c) Determine the force that vanishes the moment it breaks loose from the
hemisphere. [1 m]
PSPM 2008/2009 SF017/2 No. 11(c)
15.
0.6 m
0.5 m 0.5 m
FIGURE 6.6
FIGURE 6.6 shows a 0.3 kg object attached to a horizontal rod by two strings, each of
-1
length 0.5 m. The object is whirled with speed 8.0 m s around the axis of the rod.
Calculate the tension in the string when the object is at the
(a) lowest point. [5 m]
(b) highest point. [5 m]
4
PSPM SP015
PSPM 2010/2011 SF017/2 No. 3
16. A 0.2 kg ball, attached to the end of a string, is rotated in a horizontal circle of radius
1.5 m on a frictionless table surface. The string will snap when the tension exceeds
50 N.
(a) What is the maximum speed of the ball? [2 m]
(b) If there were friction on the table, what will happen to the maximum speed of the
ball? Explain your answer. [2 m]
PSPM 2010/2011 SF017/2 No. 11
17.
A O
L = 2 m
B
FIGURE 6.7
A small bob of mass 10 g is attached to the end of a massless string of length 2 m. The
other end of the string is fixed at point O as shown in FIGURE 6.7. Initially, the bob is
held at point A which is at the same level as point O, keeping the string taut and then
released. Determine
(a) the linear velocity of the bob at point B directly beneath point O. [3 m]
(b) the angular velocity of the bob at point B. [2 m]
(c) the tension of the string when the bob is at point B. [4 m]
(d) the linear velocity of the bob at a point midway through arc AB. [3 m]
(e) whether the linear velocity of the bob at point B be lower, higher or similar if the
experiment is performed on the surface of the moon. Explain your answer briefly.
[3 m]
PSPM 2011/2012 SF016/2 No. 4(b)
-1
18. A 3500 kg car enters a curve of radius 10 m on a flat road at 30 m s .
(a) Calculate the net horizontal force required to keep the car moving around the
curve without slipping. [2 m]
(b) State the force that enables the car to successfully negotiate the curve. [1 m]
5
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2012/2013 SF016/2 No. 4(b)
19.
30
r
FIGURE 6.8
FIGURE 6.8 shows a bob revolves in a horizontal circle of radius r. The string has a
length of 0.5 m and makes 30 with the vertical.
(a) Sketch the free body diagram for the bob. [1 m]
(b) Calculate the speed of the bob. [5 m]
(c) Calculate the period of revolution. [2 m]
PSPM 2013/2014 SF016/2 No. 4(b)
20. A 60 cm conical pendulum bob revolves freely. The pendulum string makes an angle of
37 with the vertical.
(a) Sketch and label a free body diagram of the pendulum bob. [2 m]
(b) Calculate the speed of the pendulum bob. [3 m]
(c) Calculate the angular velocity of the pendulum bob. [2 m]
(d) If the angle remains unchanged but a longer string is used, will the angular
velocity decrease, same or increase? Justify your answer. [2 m]
6
PSPM SP015
PSPM 2014/2015 SF016/2 No. 3(c)
21.
7 m
8 m
FIGURE 6.9
A theme park ‘loop the loop’ ride has a loop of radius 8 m. A 90 kg car goes around the
inside of the loop after descending from vertical height of 7 m above the top of the loop
as in FIGURE 6.9. Calculate
(a) the centripetal acceleration of the car, [3 m]
(b) the force exerted on the car [3 m]
when it is on the top of the loop.
PSPM 2014/2015 SF016/2 No. 4(a)
22. Two ice skaters, each of mass 60 kg hold hands and spin in a circle with a period of
4.3 s. The distance between them is 1.44 m.
(a) Calculate the angular velocity. [1 m]
(b) Calculate the centripetal acceleration. [1 m]
(c) Calculate the pulling force that acts on each hand. [1 m]
(d) Describe their subsequent motion after they separate. [1 m]
PSPM 2015/2016 SF016/2 No. 4(a)(ii), (iii)
23. (a) A cyclist rides on a horizontal circular track. With the aid of an appropriate
labeled force diagram, explain why the cyclist has to lean towards the centre of
the track. [2 m]
(b) Refer to the cyclist in question (a). Let v be the maximum speed of the cyclist, r
the radius of the track, the coefficient of friction between the tyres and track,
and g the gravitational acceleration. Determine in terms of v, r and g. [5 m]
7
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2016/2017 SF016/2 No. 4(b)
24.
FIGURE 6.10
FIGURE 6.10 shows a 250 g bob suspended by a light string rotating in a horizontal
-1
circle of radius 30 cm with a linear speed of 2 m s .
(a) Calculate the angle which the string makes with the vertical axis. [3 m]
(b) Calculate the tension in the string. [1 m]
(c) If the speed is increased, how would it affect the circular motion of the bob?
Explain your answer. [2 m]
PSPM 2017/2018 SF016/2 No. 4(c)
25.
T a
50°
50°
T b
FIGURE 6.11
FIGURE 6.11 shows a 4.0 kg object is attached to a vertical rod by two strings. The
-1
object rotates in a horizontal circle of radius 1.30 m at constant speed 6.0 m s .
Calculate the
(a) tension T a in the upper string. [6 m]
(b) tension T b in the lower string. [1 m]
8
PSPM SP015
PSPM CHAPTER 7: GRAVITATION
___________________________________________________________________________
24
6
Given mass of Earth = 5.98 10 kg and radius of Earth = 6.4 10 m
Gravitational Force
PSPM 2017/2018 SF016/2 No. 4(a)(i)
1. State Newton’s law of gravitation. [1 m]
PSPM 2009/2010 SF017/2 No. 3(a)
24
30
2. Given the masses of the sun and the earth are 1.99 × 10 kg and 5.98 × 10 kg
respectively. Calculate the gravitational force between them when their centres are
11
1.50 × 10 m apart. [2 m]
PSPM 2011/2012 SF016/2 No. 4(d)
3. x
m 2 m 1 m 3
15 m
FIGURE 7.1
FIGURE 7.1 shows a mass m 1 positioned between two masses, m 2 = 40 kg and
m 3 = 60 kg. m 2 and m 3 are separated by 15 m. m 1 is placed at a distance x from m 2.
(a) Sketch and label the forces acting on mass m 1 due to m 2 and m 3. [2 m]
(b) Determine x when m 1 will experience zero net gravitational force. [2 m]
Gravitational Field Strength
PSPM 2013/2014 SF016/2 No. 4(c)(i)
4. Define gravitational field strength. [1 m]
PSPM 2014/2015 SF017/2 No. 4(c)(ii)
5. Calculate the acceleration of Earth gravity at a distance of 500 km above the Earth
24
6
surface. The radius and the mass of Earth are 6.4 10 km and 5.98 10 kg
respectively. [3 m]
9
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2015/2016 SF017/2 No. 4(b)(iii)
23
6. The radius and mass of planet Mars are 3390 km and 6.42 10 kg respectively.
Calculate the acceleration due to gravity at the surface of the planet. [2 m]
PSPM JUN 1999/2000 SF015/2 No. 5
-3
7. Density and radius of Earth are 5500 kg m and 6400 km respectively. By using
suitable equation, determine the gravitational acceleration on the surface of Earth.
[4 m]
PSPM 2007/2008 SF017/2 No. 3
8. At what altitude from the surface of the earth will the acceleration due to gravity be
6
reduced by 10 percent? Radius of the earth = 6.4 × 10 m. [4 m]
PSPM JUN 1999/2000 SF015/2 No. 12(b)
9. (a) Explain qualitatively why the weight of a body on moon is less than its weight on
the surface of Earth. [4 m]
(b) Given radius of Earth and moon are 6399 km and 1782 km respectively.
Determine the relation between mass of Earth and moon if the weight of a body
on moon is 0.16 times its weight on Earth. [3 m]
PSPM 2017/2018 SF016/2 No. 4(b)(ii)
10. The international Space Station (ISS) is launched from the Earth and orbiting at the
6
altitude of 350 km. The ISS has a weight of 4.22 10 N when it is measured on the
24
surface of the Earth. Given the mass and radius of the Earth is 5.98 10 kg and
6
6.38 10 m respectively, calculate the weight of ISS when in its orbit. [2 m]
Gravitational Potential Energy
PSPM 2002/2003 SF017/2 No. 3
6
11. A satellite of mass 50 kg moves in an orbit at the height of 2 10 m from the surface
of Earth. Calculate
(a) the potential energy of Earth satellite system. [2 m]
(b) the gravitational acceleration exerted on the satellite. [2 m]
6
24
(Given mass of Earth = 5.98 10 kg and radius of Earth = 6.37 10 m)
10
PSPM SP015
Satellite motion in a circular orbit
PSPM JAN 1999/2000 SF015/2 No. 7
12. A spaceship near the surface of Earth needs a certain speed so that it can orbit Earth.
Calculate the speed of the satellite and the time for it to make one circle around Earth.
[3 m]
PSPM JAN 1999/2000 SF015/2 No. 12(b)
13. A satellite is at a distance r from the centre of Earth. By applying Newton’s
gravitational force, determine the velocity v of the satellite motion and then determine
the period of the satellite orbiting Earth in terms of r, mass of Earth M, and gravitational
constant G. [4 m]
PSPM JAN 1999/2000 SF015/2 No. 12(c)(ii)
14. Jupiter planet has radius 11.2 times of the radius of Earth and mass 318 times of the
mass of Earth. If the rotation period of this planet on its axis is 10.2 hours, calculate the
orbital radius of the satellite that orbits Jupiter.
7
(Given orbital radius of satellite that orbits Earth = 4.42 10 m) [4 m]
PSPM JUN 2000/2002 SF015/2 No. 5
15. A satellite orbits Earth in a circular orbit at the height of 35 000 km from the surface of
Earth. Calculate the velocity of the satellite. [4 m]
PSPM 2003/2004 SF017/2 No. 11(c)(ii)
16. Calculate the speed of the Hubble space telescope that orbits Earth at the height of
598 km from the surface of Earth. [3 m]
PSPM 2005/2006 SF017/2 No. 11(b)(i) – (iv) Edited
17. MATSAT is a 2000 kg satellite that is seen stationary to an observer on Earth. It is
orbiting with radius r = 42000 km from the centre of Earth.
(a) Why does the satellite appear stationary to the observer on Earth? [1 m]
(b) Explain why the satellite accelerates whereas it is moving at constant speed.
[1 m]
(c) Calculate the acceleration of the satellite. [3 m]
(d) Calculate the escape velocity of the satellite from the atmosphere of Earth. [3 m]
11
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2006/2007 SF017/2 No. 11(c)(i), (iii)
3
15
18. A spherical asteroid has a radius of 12 10 m and a mass of 4 10 kg.
(a) Calculate the gravitational acceleration at the surface of the asteroid. [3 m]
(b) Suppose the asteroid spins about an axis through its centre with an angular speed
. What is the value of when a loose rock at the equator of the asteroid begins
to fly off its surface? [3 m]
PSPM 2011/2012 SF016/2 No. 4(c)
19. A 1000 kg satellite is in an orbit 350 km above the earth surface. Calculate the
(a) speed of the satellite. [2 m]
(b) period of the satellite. [2 m]
(c) radial acceleration of the satellite. [2 m]
24
6
Given mass of Earth = 5.98 10 kg and radius of Earth = 6.4 10 m
PSPM 2013/2014 SF016/2 No. 4(c)(ii)
24
20. The mass and radius of the earth are 5.974 10 kg and 6371 km respectively.
Calculate the period of revolution of a satellite that is 100 km above the earth surface.
[2 m]
PSPM 2014/2015 SF017/2 No. 4(c)(iii)
21. A satellite orbits Earth in a period of 24 hours so that it is always at the same point
above Earth surface. Calculate the radius of the orbit. The radius and the mass of Earth
24
6
are 6.4 10 km and 5.98 10 kg respectively. [3 m]
PSPM 2015/2016 SF016/2 No. 4(c)
22. (a) Derive the period of a satellite motion. [2 m]
(b) Calculate the distance from the centre of Earth for a satellite to remain on the
24
same location above the Earth equator. The mass of Earth is 5.97 10 kg.
[1 m]
PSPM 2016/2017 SF016/2 No. 4(c)(iii)
7
23. The period of a moon orbiting Pluto is 4.8 days. The radius of the orbit is 4 10 m.
Calculate the mass of Pluto. [7 m]
12
PSPM SP015
PSPM CHAPTER 8: ROTATION OF RIGID BODY
___________________________________________________________________________
Static Equilibrium of a Uniform Rigid Body
PSPM 2006/2007 SF017/2 No. 11(a)
1. Without using any equation, explain what is meant by torque. [1 m]
PSPM JUN 1999/2000 SF015/2 No. 10(a)(ii)
2. A uniform wood of 3.0 m long has mass of 10 kg is carried by two man of the same
height. Determine the forces felt by both men if both of them support the wood at 0.6 m
and 0.8 m from the ends of the wood respectively. [4 m]
PSPM JAN 2000/2001 SF015/2 No. 10(c)
3. L
x
F student 40 N
60 N
FIGURE 8.1
A student carries a uniform rod of 30 N and length L. Two loads 40 N and 60 N is hung
at both ends as shown in FIGURE 8.1.
(a) Draw FIGURE 8.1 and show accurately the position and the direction of weight
of the uniform rod. [1 m]
(b) Calculate the value of x so that the system is in equilibrium. [5 m]
PSPM 2002/2003 SF017/2 No. 12(b)
4. A B
FIGURE 8.2
A uniform metal rod of length 1.0 m is placed with both ends both on weighing
machines as shown in FIGURE 8.2. The weight of the metal is 3 N.
(a) Draw the forces acting on the metal rod in free body diagram. [2 m]
(b) Calculate the reading of each weighing machine. [2 m]
13
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2004/2005 SF017/2 No. 11(c)
5.
C
60
cable
A rod B
hinge
70 cm 50 cm
1000 N
FIGURE 8.3
FIGURE 8.3 shows a 1000 N load hung from a 1.2 m uniform rod of weight 500 N.
The horizontal rod is pivoted to the wall by hinge A and supported by cable BC.
(a) Calculate the tension in cable BC. [2 m]
(b) Determine the magnitude and direction of the force on A. [4 m]
PSPM 2005/2006 SF017/2 No. 3
6.
R
P 12 Q
S
2 1
3 L 3 L
600 N
FIGURE 8.4
A 250 N uniform rod is hinged horizontally to a wall at P by a string QR. A 600 N load
is suspended at S as shown in FIGURE 8.4. If the system is in equilibrium, calculate
the tension in string QR. [4 m]
14
PSPM SP015
PSPM 2006/2007 SF017/2 No. 11(b)
7.
0.3 m
2.4 m
FIGURE 8.5
FIGURE 8.5 shows a uniform plank, 4.0 m long and mass 6.0 kg, supported by two
sawhorses, at 0.3 m and 2.4 m from its right end. A 2.5 kg cat walks on it from right to
left until a certain position where the plank just begins to tip.
(a) Draw all the forces acting on the plank when it just begins to tip. [4 m]
(b) Calculate the position of the cat from the left sawhorse when the plank just begins
to tip. [2 m]
PSPM 2007/2008 SF017/2 No. 2(a)
8.
ladder smooth
wall
rough floor
FIGURE 8.6
A uniform ladder is at rest on a rough floor and leaning against a smooth wall as shown
in FIGURE 8.6. Sketch a free body diagram showing THREE forces acting on the
ladder. [3 m]
15
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2010/2011 SF017/2 No. 2
9. 120 cm
15 cm
750 N
FIGURE 8.7
A 750 N athlete is doing a push-up exercise by putting his palms and toes on the floor
as shown in FIGURE 8.7. The athlete’s centre of mass is 15 cm from his palms. The
horizontal distance between his toes and palms is 120 cm. Assume the posture of the
athlete as a rigid body,
(a) draw a free body diagram of the athlete. [1 m]
(b) calculate the force acting on each of his palms. [3 m]
PSPM 2014/2015 SF016/2 No. 5(b)
10.
R Q
P
30
FIGURE 8.8
FIGURE 8.8 shows a crane with 8 m uniform arm PQ of mass 2400 kg pivoted at P.
The arm is supported by a horizontal cable QR.
(a) Sketch and label all the forces acting on the arm PQ. [3 m]
(b) Calculate the tension in cable QR. [3 m]
(c) If Q is raised by shortening cable QR, what is the effect to the tension of the
cable? Explain your answer. [2 m]
16
PSPM SP015
PSPM 2015/2016 SF016/2 No. 5(a)
11. P
Q min
FIGURE 8.9
FIGURE 8.9 shows a uniform beam PQ of weight 240 N leaning on a smooth wall and
resting on a rough floor with a minimum inclination angle min. The coefficient of
friction between the beam and the floor is 0.25.
(a) State two (2) conditions for equilibrium of the beam. [1 m]
(b) Sketch all forces acting on the beam. [2 m]
(c) Determine the forces acting on the beam at P and Q. [5 m]
(d) Determine min. [2 m]
(e) What will happen to the beam if the inclination angle is changed? Consider both
cases new < min and new > min with new less than 90. [2 m]
Rotational Kinematics
PSPM 2013/2014 SF016/2 No. 5(a)(i)
12. Define angular acceleration. [1 m]
PSPM 2016/2017 SF016/2 No. 5(a)(i)
13. Define instantaneous angular acceleration. [1 m]
PSPM JAN 2000/2001 SF015/2 No. 12(b)
-1
14. A pulley of radius 8.0 cm is connected to a motor that rotates at a rate of 7000 rad s
-1
and then decelerate uniformly at a rate of 2000 rad s within 5 s.
(a) Calculate its angular acceleration. [2 m]
(b) What is the number of rotations within the time range? [2 m]
(c) How long is the string that winds it within the time range? [2 m]
(d) Determine the tangential acceleration of the string. [2 m]
(e) Calculate the time taken for the pulley to stop. [2 m]
17
PHYSICS PSPM SEM 1 1999 - 2017
PSPM JAN 2000/2001 SF025/2 No. 1
-1
15. A bicycle wheel rotates with an angular velocity of 10π rad s . At 3 seconds after that,
-1
its angular velocity becomes 4π rad s . Calculate
(a) the angular acceleration of the wheel. [2 m]
(b) the time taken for the wheel to stop. [1 m]
PSPM JAN 2000/2001 SF025/2 No. 9(b)
16. A light string is wound the outer surface of a disc of diameter 40 cm. This disc is free to
rotate about an axis at the centre of the disc. The string is then pulled with an
-2
acceleration of 1.2 m s for 1.8 s.
(a) Calculate the angular speed of the disc at its final period of acceleration. [3 m]
(b) What is the total rotation of the disc through the period of acceleration? [3 m]
PSPM JUN 2000/2002 SF025/2 No. 1
17. A barrel of a washing machine makes 1200 rotations per minute. If the diameter of the
barrel is 35 cm, calculate
(a) the rotating frequency in Hz. [2 m]
(b) the angular velocity of the barrel. [2 m]
(c) the linear speed at the rim of the barrel. [2 m]
PSPM 2004/2005 SF017/2 No. 3
18. A wheel turns about its axis of rotation at uniform angular acceleration from rest to
-1
reach an angular velocity of 30 rad s in 20 s. Calculate
(a) its angular displacement after 4 s. [2 m]
(b) its tangential acceleration at a point 12 cm from the axis of rotation. [2 m]
PSPM 2006/2007 SF017/2 No. 3
-1
19. A spinning top has an initial angular velocity of 35 rad s . After 0.6 s, the angular
-1
velocity decreases to 33 rad s . Assuming a uniform angular deceleration, calculate
(a) the angular deceleration. [2 m]
(b) the number of revolutions during the 0.6 s interval. [2 m]
18
PSPM SP015
PSPM 2013/2014 SF016/2 No. 5(b)
20. A ceiling fan is rotating at an angular speed of 300 rpm when the switch is turned off. It
takes 45 s for the fan to stop. Calculate the
(a) average angular acceleration. [2 m]
(b) number of revolution the fan makes before it stops. [2 m]
PSPM 2014/2015 SF016/2 No. 5(a)(ii)
-1
21. At the instant a ceiling fan is switched off, the angular velocity of the fan is 65 rad s
and stop after 9 s. Calculate the number of rotation for the fan to stop completely.
[2 m]
PSPM 2017/2018 SF016/2 No. 5(b)
22. A disc of 8.0 cm in radius rotates at a constant rate of 1200 revolution per minute about
its central axis. Calculate the
-1
(a) angular velocity in rad s . [2 m]
(b) tangential speed at a point 3.0 cm from its centre. [2 m]
(c) Distance travelled by a point at the edge of the disc in 2.0 s. [2 m]
Rotational Dynamics
PSPM 2013/2014 SF016/2 No. 5(a)(ii)
23. Define moment of inertia. [1 m]
PSPM 2011/2012 SF016/2 No. 5(a)
24. State three (3) factors which affect the moment of inertia of a rigid body. [2 m]
19
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2007/2008 SF017/2 No. 11(a)
25. Given that s, v, a, and F are parameters associated with linear motion. Copy and
complete TABLE 8.1 with their corresponding parameters in rotational motion and
their relationships.
TABLE 8.1
Linear Rotational Relationship
s
v
a
F
[2 m]
PSPM 2008/2009 SF017/2 No. 3(b)
26. y (m)
4
3 8.0 kg
2
1
5.0 kg P 7.0 kg
x (m)
1 2 3 4 5
FIGURE 8.10
A three-body system is arranged as shown in FIGURE 8.10. If the axis of rotation is
perpendicular to the plane of the paper through point P, calculate the moment of inertia
of the system. [1 m]
PSPM JAN 1999/2000 SF025/2 No. 10(b)(ii)
2
27. A solid cylinder has moment of inertia MR at the centre of mass of the cylinder
1
2
which has mass M and radius R. If the cylinder rolls with linear velocity v and angular
velocity ω, show that the energy of the rolling cylinder is given by
2
K = Mv [3 m]
3
4
20
PSPM SP015
PSPM JAN 1999/2000 SF025/2 No. 10(c)
28. A solid sphere and a ring shaped body having the same mass M and radius R are let to
roll in the same direction on a flat plane. Calculate the ratio of kinetic energy of the ring
to the kinetic energy of the solid sphere.
2
2
(Given moment of inertia of sphere = MR and moment of inertia of ring = MR )
2
5
[5 m]
PSPM JUN 1999/2000 SF025/2 No. 1
29. A cylindrical roller of mass 2.8 kg rotates with a speed of 1500 rotations per minute. If
the diameter of the cross-sectional area is 32.0 cm,
(a) calculate the angular momentum of the cylindrical roller. [3 m]
(b) determine the torque required to stop the cylindrical roller in 7.0 s. [1 m]
PSPM 2016/2017 SF016/2 No. 5(c)
30.
A
60
50 cm B
FIGURE 8.11
FIGURE 8.11 shows a tiny ball with mass, M is released from point A rolls without
slipping on the inside surface of a hemisphere with radius of curvature 50 cm. The
2
moment of inertia of the ball is MR where R is the radius of the ball. Calculate the
2
5
speed of the ball at point B. [4 m]
PSPM JAN 2000/2001 SF025/2 No. 9(c)
31.
2 m s -1
h
FIGURE 8.12
-1
A solid sphere rolls with initial velocity of 2 m s as shown in FIGURE 8.12. If the
energy loss through friction is negligible, determine the maximum height that can be
achieved. [6 m]
21
PHYSICS PSPM SEM 1 1999 - 2017
PSPM JUN 2000/2002 SF025/2 No. 9(c)
32. Two identical balls 0.5 kg each is tied to both ends of a light rod of 1.0 m long. A
member of a circus held the rod at the centre and rotates it.
(a) What is the moment of inertia of the ball system about its axis of rotation? [3 m]
(b) If the rod is rotated 30 rotations per minute, what is the angular momentum?
[2 m]
(c) Determine the kinetic energy of the system. [2 m]
PSPM 2001/2002 SF015/2 No. 11(b)(i), (iii)
33.
rotation axis
marble R track of cone
60
FIGURE 8.13
A small track of conical shape can rotate about its axis has angle of 60 as shown in
FIGURE 8.13. A marble of mass m is placed on the track. When the track rotates with
an angular velocity of , the marble does not move at a distance R.
(a) Draw all the forces that act on the marble. [3 m]
(b) If the cone stops rotating, the marble will roll down. Calculate the acceleration of
the marble while going down the track of the cone. Given moment of inertia of the
2
2
marble is mr where r is the radius of the marble. [5 m]
5
PSPM 2002/2003 SF017/2 No. 12(c)
2
34. A disc of moment inertia 0.05 kg m is fixed on an axis of rotation that rotates at a rate
of 30 rotations per second. Then its motion is slowed down uniformly until it stops. If
the disc rotates 60 rotations before it stops, calculate
(a) the angular acceleration of the disc. [3 m]
(b) the time taken for the disc to stop. [3 m]
(c) the change of kinetic energy of the disc rotation. [3 m]
22
PSPM SP015
PSPM 2004/2005 SF017/2 No. 11(b)
35.
0.2 m
50 kg cable
9 N
FIGURE 8.14
FIGURE 8.14 shows a massless cable wound around a solid cylinder of mass 50 kg
and radius 0.2 m. When the free end of the cable is pulled through a distance of 2.0 m
by a constant force of 9 N, the cylinder initially at rest, rotates without friction about a
stationary horizontal axis. Calculate
(a) the final angular velocity of the cylinder. 5 m]
(b) the final speed of the cable. [2 m]
2
1
(Moment inertia of a solid cylinder of mass M and radius R is I = MR )
2
PSPM 2007/2008 SF017/2 No. 11(b), (c)
36. (a)
A 10 m
30
B
FIGURE 8.15
A solid cylinder of radius 30 cm and mass 2 kg starts from rest at A and rolls
without slipping down a plane inclined 30 to the horizontal. The cylinder
traverses a distance of 10 m along the plane to reach B as shown in
FIGURE 8.15. Calculate
(i) the potential energy of the cylinder at A. [2 m]
(ii) the linear velocity of the cylinder just before reaching B. [5 m]
(iii) the time taken for the cylinder to reach B. [3 m]
(b) If a hollow cylinder of similar mass and radius are released simultaneously with
the cylinder from A as in question (a), which cylinder will be the fastest to reach
B? Explain your answer. [3 m]
2
Given the moments of inertia of a solid cylinder and a hollow cylinder is 1 2 MR and
2
MR respectively.
23
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2009/2010 SF017/2 No. 11
37.
M
R
m
FIGURE 8.16
FIGURE 8.16 shows a uniform solid cylinder of mass M = 2.0 kg and radius R = 10 cm
pivoted horizontally so that it is free to rotate. A mass m = 3.0 kg is suspended by a
string that is wound around the cylinder. The system is released from rest. Given the
2
moment of inertia of the solid cylinder, I 1 2 MR
(a) Calculate the acceleration of the mass m. [6 m]
(b) Calculate the tension in the string. [2 m]
(c) Upon completion of 5 revolutions by the cylinder, calculate the
(i) rotational kinetic energy of the cylinder. [5 m]
(ii) angular momentum of the cylinder. [2 m]
PSPM 2011/2012 SF016/2 No. 5(b)
38.
1.5 kg
FIGURE 8.17
FIGURE 8.17 shows a 1.5 kg block hung by a light string which is wound around a
2
pulley of radius 20.0 cm. The moment of inertia of the pulley is 2.0 kg m . When the
mass is released from rest, calculate the
(a) torque exerted on the pulley. [4 m]
(b) angular velocity of the pulley at t = 4.2 s. [1 m]
(c) number of revolutions made by the pulley in 4.2 s. [2 m]
24
PSPM SP015
PSPM 2011/2012 SF016/2 No. 5(c)
39.
P
h
Q
FIGURE 8.18
A solid sphere and a hollow sphere have the same mass, m and radius, r. FIGURE 8.18
shows both spheres are released simultaneously from rest at P and roll down on an
inclined plane. Which sphere will arrive first at Q? Show your answer. (Moment of
2
2
inertia of the spheres: I solid 2 5 mr and I hollow 2 3 mr ) [6 m]
PSPM 2012/2013 SF016/2 No. 5(b)
40. A disc rotates at 25 revolutions per minute and takes 15 s to come to rest. Calculate
(a) the angular acceleration of the disc. [2 m]
(b) the number of rotation the disc makes before it comes to rest. [2 m]
2
(c) the initial kinetic energy of the disc if its moment of inertia is 2.5 kg m . [3 m]
PSPM 2012/2013 SF016/2 No. 5(c)
41.
2 kg
FIGURE 8.19
FIGURE 8.19 shows a 2 kg box hung by a cord wound on a light pulley is released
2
from rest. The radius and the moment of inertia of the pulley are 0.25 m and 1.5 kg m .
Calculate the acceleration of the box. [6 m]
25
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2013/2014 SF016/2 No. 5(c)
42.
32 cm
4000 rpm
6 cm
FIGURE 8.20
FIGURE 8.20 shows a 2.76 kg solid disc of diameter 32 cm with a 2.66 kg solid
cylindrical axle of diameter 6 cm. The moment of inertia of the hollow disc is
2
1 M (R R 2 ) where R 1 is the radius of solid disc and R 2 is the radius of cylindrical
2 disc 1 2
2
1
axle. The moment of inertia of the cylinder is M cylinder R . Calculate the
2
2
(a) moment of inertia of the system about the axis of the cylindrical axle. [2 m]
(b) energy required to rotate the system about the axis from an angular speed of
1000 rpm to 4000 rpm. [3 m]
PSPM 2013/2014 SF016/2 No. 5(d)
43. A solid sphere and a solid cylinder from rest roll without slipping down a slope from
the same height. Both of them have the same mass and radius.
Given the moment of inertia of the solid sphere and solid cylinder are 2 5 MR and
2
2
1 MR respectively, which one of them will reach the bottom of the slope first? Justify
2
your answer.
[3 m]
PSPM 2014/2015 SF016/2 No. 4(b)
44. Two ice skaters, each of mass 60 kg hold hands and spin in a circle with a period of
4.3 s. The distance between them is 1.44 m.
(a) Calculate the angular velocity. [1 m]
(b) Calculate the centripetal acceleration. [1 m]
(c) Calculate the pulling force that acts on each hand. [1 m]
(d) Describe their subsequent motion after they separate. [1 m]
26
PSPM SP015
PSPM 2015/2016 SF016/2 No. 5(b)
45. A 15 N m torque acts on a grinding wheel after the power is switched off. Given the
2
angular velocity of the wheel is 1200 rpm and its moment of inertia is 0.8 kg m ,
calculate the time for the wheel to stop. [3 m]
PSPM 2016/2017 SF016/2 No. 5(b)
2
46. A flywheel with moment of inertia 8 kg m is acted upon by a constant torque of
50 N m. Calculate the
(a) angular acceleration. [1 m]
-1
(b) time for it to rotate from rest to 70 rad s . [1 m]
-1
(c) power when the angular velocity is 50 rad s . [1 m]
PSPM 2017/2018 SF016/2 No. 5(c)
47.
1.5 m
F = 0.8 N
FIGURE 8.21
FIGURE 8.21 shows a 0.75 kg airplane model is attached by an arm rod of negligible
mass flies in horizontal circle 1.5 m in radius. The airplane model acts a force of 0.8 N
perpendicular to the rod. By assuming the airplane as a point mass, calculate the
(a) torque about the axis passing through the centre of the circle. [2 m]
(b) angular acceleration of the airplane. [2 m]
(c) linear acceleration of the airplane tangent to its flight path. [2 m]
27
PHYSICS PSPM SEM 1 1999 - 2017
Principle of Conservation of Angular Momentum
PSPM 2017/2018 SF016/2 No. 5(a)(ii)
48. Define angular momentum. [1 m]
PSPM JUN 1999/2000 SF025/2 No. 9(a)
49. State the principle of conservation of angular momentum. [2 m]
PSPM JAN 1999/2000 SF025/2 No. 10(d)
50. A raw egg and a boiled egg are let to rotate initially on a similar axis with the same
angular velocity. Explain which egg will rotate longer. [3 m]
PSPM JUN 1999/2000 SF025/2 No. 9(b)
51.
axis of rotation
2.0 m O X
1.2 m
F
platform
FIGURE 8.22
FIGURE 8.22 shows a 1.6 kg cat X at 1.2 m apart from the axis of rotation through
point O on the platform that rotates horizontally without friction. The platform has
2
moment of inertia 80.0 kg m and radius 2.0 m. It is rotating under a tangential force F
of constant magnitude 25.0 N.
(a) What is the moment of inertia for the whole rotating system? [2 m]
(b) If the platform starts to rotate from rest, what is the angular momentum of the
system after 9.0 s? [5 m]
(c) Calculate the work done on the platform in 9.0 s. [3 m]
(d) After 9.0 s, force F is taken out so that the platform and the cat rotate with a
constant angular velocity. What is the change that occurs to the platform if the cat
moves towards point O? Give reason to your answer. [3 m]
28
PSPM SP015
PSPM 2014/2015 SF016/2 No. 5(c)
52.
FIGURE
8.23
FIGURE 8.23 shows a ball attached to one end of a string which is swung in a
-1
horizontal circle of radius 60 cm with an initial speed of 3 m s . The other end of the
string passes through a tube. If the string is slowly pulled downwards until the speed of
-1
the ball becomes 5 m s , what is the radius of the circular path of the ball?
[3 m]
PSPM 2016/2017 SF016/2 No. 5(d)
53.
ball
5 m s -1
pivot
1 m
FIGURE 8.24
FIGURE 8.24 shows a top view of a 1 m rod pivoted freely at its center and placed
-1
horizontally. A 20 g ball with velocity 5 m s collides and sticks at one end of the rod
2
causing them to rotate. The moment of inertia of the ball-rod system is 0.02 kg m .
Calculate the angular velocity of the rod. [4 m]
29
PHYSICS PSPM SEM 1 1999 - 2017
PSPM CHAPTER 9: SIMPLE HARMONIC MOTION
___________________________________________________________________________
Kinematics of Simple Harmonic Motion
PSPM 2016/2017 SF016/2 No. 6(a)
1. (a) What is meant by simple harmonic motion? [1 m]
(b) Is a bouncing ball an example of simple harmonic motion? Explain your answer.
[1 m]
PSPM 2013/2014 SF016/2 No. 6(a)
2. The displacement, x of a simple harmonic motion given by the equation,
d 2 x 2 x
dt 2
where
is a constant associated with the motion. What is meant by the equation?
[1 m]
PSPM 2017/2018 SF016/2 No. 6(a)
3. Why simple harmonic motion is a periodic motion without energy loss? [1 m]
PSPM 2014/2015 SF016/2 No. 6(a)
4. State TWO properties of an object undergoing simple harmonic motion. [2 m]
PSPM 2008/2009 SF017/2 No. 12(a)
5. State the relationship between simple harmonic motion and uniform circular motion.
[2 m]
PSPM 2008/2009 SF017/2 No. 12(b)
6. What are the characteristics of the force in a simple harmonic motion? [2 m]
PSPM 2010/2011 SF017/2 No. 4(a)
7. State one difference of the accelerations between linear motion and that of simple
harmonic motion. [1 m]
30
PSPM SP015
PSPM JAN 1999/2000 SF035/2 No. 1(b)
8. A mass is hung at the end of a spring. The mass is pulled down and is released so that it
oscillates vertically. The distance between upper point and lower point is 0.4 m. If the
mass takes 0.75 s to travel this distance, calculate
(a) the angular speed. [1 m]
(b) the maximum speed. [2 m]
PSPM JAN 1999/2000 SF035/2 No. 9(e)
9. A 50 g mass is connected to a spring that moves horizontally on a smooth surface in
simple harmonic motion with amplitude 16 cm and period 4 s. Determine the total
mechanical energy of the system and its spring constant. [3 m]
PSPM JUN 2000/2002 SF035/2 No. 9(b)(i), (ii), (iii)
10. A 0.5 kg body on one end of a spring oscillates with 3.0 cycles per second. If the
amplitude of oscillation is 0.15 m,
(a) calculate the speed of the body when it passes the equilibrium point. [2 m]
(b) calculate the speed of the body when it is at 0.1 m from the equilibrium point.
[2 m]
(c) determine the total energy of the system. [2 m]
PSPM 2005/2006 SF017/2 No. 4
11. A body oscillates with simple harmonic motion described by the following expression
y = 3 sin 4t
where y and t are displacement in meter and time in second, respectively.
(a) Calculate the oscillating period. [1 m]
(b) At t = 2 s,
(i) calculate the speed of the body. [2 m]
(ii) state the direction of body motion. [1 m]
31
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2006/2007 SF017/2 No. 4
12. An object is executing a simple harmonic motion with an amplitude of 30 m and a
-2
maximum acceleration of 15 m s . Calculate
(a) the period of motion. [2 m]
(b) the maximum speed of the object. [2 m]
PSPM 2008/2009 SF017/2 No. 4
13. A 0.9 kg object performs a simple harmonic motion according to the following equation
y cos 3t
4
2
where y is in meter and t is in second. At time t = 7 s, calculate the
(a) acceleration of the object. [2 m]
(b) force on the object. [2 m]
PSPM 2008/2009 SF017/2 No. 12(d)
14. At a particular instant, the potential energy of a simple harmonic motion contributes
75% of its total energy. Calculate the displacement if the amplitude of the motion is
5.0 cm. [3 m]
PSPM 2011/2012 SF016/2 No. 6(c)
15. A 200 g object is connected to a spring and moves along the y-axis in a simple
harmonic motion about the origin. The position of the object is given by
y = 0.5 cos (0.4t – 0.25)
where y is in centimeter and t is in seconds. Calculate the
(a) frequency of the motion. [2 m]
(b) velocity of the object at t = 2 s. [2 m]
(c) acceleration of the object at t = 2 s. [2 m]
(d) total energy of the system. [3 m]
32
PSPM SP015
PSPM 2013/2014 SF016/2 No. 6(b)
16. X O Y
P
5 cm
25 cm
FIGURE 9.1
FIGURE 9.1 shows a bead executing a simple harmonic motion with a period of 1.8 s,
along a straight line between points, X and Y which are 25 cm apart. Point O is at
midpoint between X and Y.
(a) Write an equation for the displacement of the bead. [4 m]
(b) Calculate the magnitude of acceleration and velocity of the bead at point P.
[2 m]
(c) Calculate the positions along XY when the kinetic energy and the potential energy
of the bead are equal. [3 m]
PSPM 2015/2016 SF016/2 No. 6(c)
17. The displacement x of a particle varies with time, t is given by
.
4
x 0 cos 2 t
2
where x is in cm and t in s. Calculate the
(a) frequency of the motion. [1 m]
(b) velocity of the particle at t = 2 s. [2 m]
(c) acceleration of the particle at t = 2 s. [2 m]
PSPM 2016/2017 SF016/2 No. 6(b)
18. A particle in simple harmonic motion along the x-axis starts from the equilibrium
position and moves towards the right with an amplitude of 2 cm and a frequency of
1.5 Hz.
(a) Write the simple harmonic motion equation of the particle. [1 m]
(b) Calculate the maximum speed of the particle. [2 m]
(c) Calculate the acceleration of the particle at t = 3.2 s. [2 m]
33
PHYSICS PSPM SEM 1 1999 - 2017
Graphs of Simple Harmonic Motion
PSPM JUN 1999/2000 SF015/2 No. 11(c) Edited
19. U (J)
0.16
A A’
B 0.04 B’
y (m)
–6 –3 0 3 6
FIGURE 9.2
FIGURE 9.2 shows the graph of potential energy against displacement of a simple
pendulum that oscillates about the equilibrium point O. By neglecting air friction,
calculate the speed at A, A’, B, B’, and O if the mass of the pendulum is 0.2 kg and the
total energy is 0.25 J by neglecting air resistance. Determine the maximum height that
can be achieved by the pendulum. [7 m]
PSPM JUN 1999/2000 SF035/2 No. 1
20.
displacement, y (cm)
2
0 time, t (s)
2 4
–2
FIGURE 9.3
FIGURE 9.3 shows a displacement against time graph of a body that moves in simple
harmonic motion.
(a) Determine the amplitude and period for this motion. [2 m]
(b) Write an equation of displacement for the motion of this body in terms of time.
[2 m]
34
PSPM SP015
PSPM JUN 2000/2002 SF035/2 No. 1
21. y (m)
3
0 t (s)
–3
FIGURE 9.4
FIGURE 9.4 shows a displacement versus time graph of a body that oscillates in
-1
simple harmonic motion at an angular speed of 2.0 rad s . Sketch and label the graph of
(a) velocity versus time [2 m]
(b) acceleration versus time [2 m]
of the body.
PSPM 2001/2002 SF015/2 No. 4
22. force, F (N)
2
0.2 displacement, y (m)
–0.2 0
–2
FIGURE 9.5
A piston of mass 0.1 kg moves in simple harmonic motion as shown in FIGURE 9.5.
Calculate
(a) the amplitude of the motion. [2 m]
(b) the angular velocity of the motion. [2 m]
PSPM 2002/2003 SF017/2 No. 4(a)(i)
23. Sketch displacement against time graph for harmonic oscillation. [1 m]
35
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2004/2005 SF017/2 No. 12(b)(i)
24. y (cm)
15
7.5
time, t (s)
0 0.1 0.5 0.9
–15
FIGURE 9.6
FIGURE 9.6 shows a displacement-time graph of simple harmonic motion (SHM).
From the graph above, calculate the amplitude and frequency.
[5 m]
PSPM 2006/2007 SF017/2 No. 12(a)(i), (ii), (iii)
25.
FIGURE 9.7
-1
A 0.5 kg block slides at 3 m s on a frictionless horizontal surface. The block hits a
spring and permanently hooked to it as shown in FIGURE 9.7. The spring constant is
-1
250 N m .
(a) Calculate the maximum compression distance of the spring. [2 m]
(b) How long is the spring in contact with the block before they come to the first
instantaneous rest? [3 m]
(c) On the same axes, sketch and label the graphs of kinetic energy, K and potential
energy, U of the block-spring system as a function of displacement, x from their
first maximum compression. [3 m]
36
PSPM SP015
PSPM 2012/2013 SF016/2 No. 6(b)
26. y (cm)
5
0 2 4 6 t (s)
–5
FIGURE 9.8
FIGURE 9.8 shows a displacement-time graph of simple harmonic motion. Determine
(a) the amplitude. [1 m]
(b) the frequency. [2 m]
(c) the angular frequency. [2 m]
(d) the equation for the simple harmonic motion. [2 m]
PSPM 2014/2015 SF016/2 No. 6(b)
27. A 25 g block connected to one end of a spring is displaced 15 cm from its equilibrim
-1
position and released. The spring constant is 180 N m .
(a) Write the acceleration a of simple harmonic motion equation of the block in terms
of displacement x. [1 m]
(b) Sketch the graph of displacement versus time of the block. Label the amplitude
and the period of oscillation on the graph. [2 m]
(c) Calculate the instantaneous speed of the block at a distance 11 cm from the
equilibrium position. [3 m]
(d) Calculate the total energy of the system. [2 m]
(e) On the same graph, sketch and label the variation of kinetic energy K and
potential U versus displacement x. On the graph, indicate the maximum value of
each curve (if any) and the amplitude. [3 m]
(f) The spring is removed and then 25 g block is connected to the a string. The string-
block system oscillates as a simple pendulum. What effect on the period of
oscillation if the mass of the block is reduced by half? Explain your answer.
[2 m]
37
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2015/2016 SF016/2 No. 6(a)
28. Sketch a graph of potential energy against displacement for SHM. [1 m]
PSPM 2015/2016 SF016/2 No. 6(b)
29. x (cm)
8
4
0 t (s)
0.2 0.4 0.6 0.8 1.0
–4
–8
FIGURE 9.9
A 0.2 kg object performs SHM with displacement, x from the equilibrium point as in
FIGURE 9.9.
(a) Determine the amplitude. [1 m]
(b) Calculate the angular velocity. [3 m]
(c) Calculate the total energy of the system. [3 m]
PSPM 2017/2018 SF016/2 No. 6(b)
30. In an engine, a 200 g piston oscillates in a simple harmonic motion with displacement
varying according to x (t) = 5 cos 2t where x is in meter and t is in second.
(a) Write the expression for its vibrational velocity as a function of time, v (t). [2 m]
(b) Sketch the graph of velocity against time of the piston for the first complete cycle
starting from t = 0 [2 m]
(c) Calculate the maximum acceleration of the system. [2 m]
(d) Calculate the total energy of the system. [2 m]
38
PSPM SP015
Period of Simple Harmonic Motion
PSPM 2002/2003 SF017/2 No. 4(b)
31. A pendulum consists of a long steel wire and a bob of mass 10 kg. In the early morning,
oscillation period is recorded as 12.0 seconds but after noon the temperature increased
and the oscillation period of the pendulum becomes 13.5 seconds. Calculate the
elongation percentage for the cable of the pendulum. [2 m]
PSPM 2003/2004 SF017/2 No. 4
32. An object of mass 100 kg suspended from a light helix spring is in simple harmonic
motion with period of 1.0 s. What additional mass should be added to the spring so that
the spring will oscillate with a period of 3.0 s? [4 m]
PSPM 2004/2005 SF017/2 No. 4
33. A simple pendulum is suspended from the ceiling of a lift. The period of oscillation T of
the pendulum when the lift is at rest is given by
T 2
g
where and g are the length and the gravitational acceleration respectively. Explain the
change in the period of the oscillation when the lift is moving
(a) upwards with acceleration a. [2 m]
(b) at constant velocity. [2 m]
PSPM 2007/2008 SF017/2 No. 4
34. A spring with a mass m attached to its end vibrates with a period of 1.5 s. When an
additional mass of 0.8 kg is added to m, the period is 2.2 s. Calculate the value of m.
[4 m]
PSPM 2008/2009 SF017/2 No. 12(c)
35. An astronaut on the moon sets up a simple pendulum of length 0.8 m. The pendulum
executes simple harmonic motion and makes 20 complete cycles in 90 s. Calculate the
acceleration due to gravity of the moon. [2 m]
PSPM 2008/2009 SF017/2 No. 4
36. A spring extends by 4 cm when a 100 g mass is suspended from it. Calculate the period
of oscillation when the mass is allowed to oscillate. [3 m]
39
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2010/2011 SF017/2 No. 4(b)
37. A mass m at the end of a spring vibrates vertically with a frequency of 0.9 Hz. When an
additional 1.2 kg is attached to m, the frequency is 0.5 Hz. Calculate the value of m.
[3 m]
PSPM 2011/2012 SF016/2 No. 6(b)
38. An object of mass 50 kg suspended from a spring undergoes a simple harmonic motion
with a period of 6.0 s.
(a) Calculate the angular velocity of the motion. [2 m]
(b) What additional mass should be added to the spring so that it oscillates with a
period of 8.0 s? [3 m]
PSPM 2012/2013 SF016/2 No. 6(c)
39. A 175 g mass on a smooth surface is attached to a horizontal spring with a spring
-1
constant 8 N m . The mass is set to oscillate by pulling it 10 cm from its equilibrium
position and released. calculate
(a) the period of the oscillation. [2 m]
(b) the maximum speed of the mass. [3 m]
(c) the total energy of the system. [2 m]
PSPM 2013/2014 SF016/2 No. 6(c)
40. A vertical spring extends by 3 cm when a 100 g mass is suspended at its end.
(a) Calculate the period of oscillation of the spring when a mass of 150 g is added to
the system. [3 m]
(b) If the spring with the same load is allowed to oscillate horizontally on a
frictionless surface, will the period decrease, same or increase? Justify your
answer. [2 m]
PSPM 2016/2017 SF016/2 No. 6(c)
-1
41. A 50 g object connected to a spring with a spring constant 35 N m oscillates with
amplitude 4 cm on a horizontal frictionless surface. Calculate the
(a) total energy of the system. [2 m]
(b) speed of the object at displacement 1.6 cm. [3 m]
(c) change in the period of oscillation if a load of 6 g is added to the object. [3 m]
PSPM 2017/2018 SF016/2 No. 6(c)
42. Explain by derivation why the period of a simple pendulum does not depend on the
mass of the bob but the period of mass-spring system depends on the mass of the load.
[6 m]
40
PSPM SP015
PSPM CHAPTER 10: MECHANICAL AND SOUND WAVES
___________________________________________________________________________
Properties of Waves ~ Progressive Wave
PSPM JAN 1999/2000 SF035/2 No. 2(a)(i)
1. A wave produced on a rope has amplitude 2.0 cm, wavelength 1.2 m, and speed
-1
6.0 m s in the positive-x direction. What is its period, frequency, angular frequency,
and wave number?
[2 m]
PSPM JAN 2000/2001 SF035/2 No. 9(a), (b)
2. (a) State three differences between stationary wave and progressive wave. [3 m]
(b) A progressive wave equation is as follows:
y = 15 sin (3t – 0.3x)
where x and y are in meter and t is in second.
Calculate the amplitude, wavelength, angular frequency, frequency, period, and
the speed of wave.
[6 m]
PSPM JUN 2000/2002 SF035/2 No. 2
3. A progressive wave is represented by equation
y = 10 sin (20t – x)
State the direction of the wave propagation. [1 m]
PSPM JUN 2000/2002 SF035/2 No. 9(c)
4. The following equation represents a wave motion:
x
y 4 sin 314t
.
0
. 0 15
where y and x are in meter and t is in second.
(a) Calculate the wavelength. [2 m]
(b) Calculate the speed of wave. [2 m]
(c) State the direction of wave propagation. [1 m]
(d) Determine the phase difference between two points at a distance of 0.45 m. [1 m]
41
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2001/2002 SF015/2 No. 5
5. A progressive wave equation is given by y = 3 sin (8t + 0.4x) cm. Determine
(a) the velocity of the wave particle v y. [1 m]
(b) the acceleration of the wave particle a y. [1 m]
(c) the speed of wave v. [2 m]
PSPM 2001/2002 SF015/2 No. 12(a)(ii)
6. A sinusoidal progressive wave moving to the right has displacement, y that equals
minimum displacement at time t = 0 s for x = 0 m. Write an equation of progressive
wave motion in terms of period T and wavelength . [4 m]
PSPM 2001/2002 SF015/2 No. 12(b)
7. displacement, y
(cm)
A
0 time, t (s)
–A
FIGURE 10.1
FIGURE 10.1 shows a progressive wave profile, y which propagates with maximum
displacement, A and angular frequency,
.
(a) Write the wave motion equation. [1 m]
(b) If a wave y 1 lags y with a phase angle of /2 rad, write the wave motion equation
for y 1. Copy FIGURE 10.1 and sketch the wave profile for y 1.
[2 m]
PSPM 2001/2002 SF015/2 No. 12(c)(i)
8. A progressive wave propagates along a string to the positive-x direction at a speed of
-1
1100 m s . If the maximum displacement is 0.5 m and does 25 cycles per second,
determine wavelength, angular frequency, and wave number. [3 m]
PSPM 2003/2004 SF017/2 No. 12(a)(i)
9. State two important properties of progressive wave. [2 m]
42
PSPM SP015
PSPM 2003/2004 SF017/2 No. 12(b)
10. Show that the equation for a progressive wave y = A sin (kx – t) can be written in the
x t
form of Ay sin 2 . [2 m]
T
PSPM 2003/2004 SF017/2 No. 12(c)
-1
11. A progressive wave with velocity 300 m s , frequency 600 Hz, and amplitude 20 cm is
propagating from left to right.
(a) Calculate the wavelength of the wave. [2 m]
(b) Write a wave equation for the wave. [4 m]
PSPM 2005/2006 SF017/2 No. 12(a)
12. Explain
(a) the speed of propagation of wave. [1 m]
(b) the vibrational speed of a particle. [1 m]
PSPM 2006/2007 SF017/2 No. 5
13. (a) What is meant by wave propagation velocity? [1 m]
(b) A transverse wave produced in a string is described by the expression,
y = 0.002 sin (30x – 2ft)
where y and x are in meter and t is in second. If the wave travels at a speed of
-1
30 m s , calculate its frequency. [3 m]
43
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2007/2008 SF017/2 No. 12(c)
14. y (cm)
2.0
0 t (s)
6
–2.0
FIGURE 10.2(a)
y (cm)
2.0
0 x (cm)
7.5
–2.0
FIGURE 10.2(b)
FIGURE 10.2(a) shows a graph of displacement, y versus time t and FIGURE 10.2(b)
shows a graph of displacement, y versus distance x of a progressive wave. From the
graphs, deduce
(a) the angular frequency. [2 m]
(b) the wave number. [2 m]
(c) the wave propagation velocity. [2 m]
(d) the particle vibrational velocity when displaced vertically 1.2 cm from the
equilibrium position.
[2 m]
(e) the displacement equation of the progressive wave. [2 m]
PSPM 2008/2009 SF017/2 No. 12(e)
-1
15. A progressive wave propagates towards positive-x direction with a velocity of 5 m s .
The amplitude and wavelength are 15 mm and 40 cm respectively.
(a) Determine the wave equation. [3 m]
(b) Calculate the vibrational speed of a particle at position x = 0.5 cm at time t = 0.2 s.
[3 m]
44
PSPM SP015
PSPM 2009/2010 SF017/2 No. 12(c)
16. A progressive wave is represented by equation
y = 20 sin (150t – 1.8x)
where y is the displacement in cm, t is the time in second and x is the distance in meter.
Calculate the wave velocity.
[3 m]
PSPM 2010/2011 SF017/2 No. 5
17. The equation of a progressive wave is given as
y = 2 sin (10t + 5x) cm
where t is in second and x is in cm. Determine the propagation velocity of the wave.
[4 m]
PSPM 2012/2013 SF016/2 No. 7(b)
18. A progressive wave is represented by the following equation:
y 10 sin t x
4
where y is in centimeter and t is in second. Determine
(a) the direction of wave propagation. [1 m]
(b) the wavelength. [3 m]
PSPM 2013/2014 SF016/2 No. 7(a)
19. A progressive wave is represented by equation,
y (x, t) = 1200 sin (314t – 0.42x)
where x and y are in cm and t is in second. Determine the
(a) velocity of the wave. [2 m]
(b) maximum velocity of the particle. [2 m]
45
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2014/2015 SF016/2 No. 7(a)(ii)
20. y (m)
0.12
0.4 0.8 1.2 1.6 x (m)
y (m)
0.12
t (s)
1 2 3 4
FIGURE 10.3
FIGURE 10.3 shows the graphs of displacement versus distance and displacement
versus time of a progressive wave travelling in the positive-x direction. Write the wave
equation. [3 m]
PSPM 2015/2016 SF016/2 No. 7(a)
21.
16 cm
4 cm
FIGURE 10.4
-1
FIGURE 10.4 shows a progressive wave travelling at a speed of 24 m s to the right.
(a) Determine the wave number. [2 m]
(b) Calculate the frequency of the wave. [2 m]
(c) Write the wave equation. [2 m]
46
PSPM SP015
Superposition of Wave ~ Standing Wave
PSPM 2016/2017 SF016/2 No. 7(a)
22. Explain the formation of stationary wave. [2 m]
PSPM JUN 1999/2000 SF035/2 No. 2(b)
23. Displacement equation of a stationary wave is given by
y = 3 sin (100t) cos (0.1x)
where y and x are in cm and t is in second. Determine the speed of this wave. [3 m]
PSPM 2007/2008 SF017/2 No. 5
-1
24. The velocity of a stationary wave on a string is 300 m s . If the frequency of the wave
is 100 Hz, what is the distance between two successive nodes? [3 m]
PSPM 2008/2009 SF017/2 No. 5
25. The following waves are superposed;
y 1 (x, t) = 6 sin (2t – 3x)
y 2 (x, t) = 6 sin (2t + 3x)
(a) What type of wave is formed? [1 m]
(b) Derive the equation of the resulting wave. [3 m]
Sound Intensity
PSPM 2013/2014 SF016/2 No. 7(c)(i)
26. Define sound intensity. [1 m]
PSPM 2010/2011 SF017/2 No. 12(b)(ii)
27. How does the intensity of sound change with the sound wave amplitude, A and distance,
r from its source? [2 m]
47
PHYSICS PSPM SEM 1 1999 - 2017
PSPM 2017/2018 SF016/2 No. 7(a)
28. (a) How does sound intensity change with distance from a point source? [1 m]
(b) In a longitudinal wave in a horizontal spring, the coils move back and forth in the
direction of wave motion. If the coil speed is increased, does the wave
propagation speed decrease, remain the same or increase? Explain your answer.
[2 m]
Application of Standing Wave ~ Stretched String
PSPM JUN 1999/2000 SF035/2 No. 9
29. A wire with both ends fixed has tension 80 N. Length and mass of the wire are 1.5 m
and 2.0 g respectively. This wire is vibrated at fundamental mode by pulling its centre
about 3 cm.
(a) Explain the difference between the type of wave formed in the wire and the wave
formed in the air due to the vibration of the wire. [2 m]
(b) Draw a diagram to show this wave and mark the nodes and antinodes. [2 m]
(c) Calculate the vibration frequency of the string at fundamental mode. [4 m]
(d) If the distance x of a particle on the wire is measured from the mid point, calculate
the amplitude of the particle at x = +0.5 m. [4 m]
(e) Calculate the displacement at x = 0.5 m at time t = 0.025 s. [3 m]
PSPM 2003/2004 SF017/2 No. 12(d)
30. A string of mass 240 g and length 2.0 m has one of its ends fixed to a wall and the other
end is forced to vibrate transversely with a frequency of 50 Hz. If the string is stretched
to a tension of 140 N, calculate the velocity and wavelength of the wave produced.
[4 m]
PSPM 2005/2006 SF017/2 No. 5(a)
31. A 2 m taut string is plucked at its centre producing vibrations with fundamental
frequency of 40 Hz. Calculate the velocity of the transverse wave propagating in the
string. [3 m]
PSPM 2009/2010 SF017/2 No. 12(b)
32. A rope of mass 0.8 kg is stretched between two supports 20.0 m apart. The tension in
the rope is 100 N. Calculate the
(a) mass per unit length of the rope. [1 m]
(b) time taken for a pulse to travel from one end to the other. [3 m]
48
PSPM SP015
PSPM 2010/2011 SF017/2 No. 12(a)(i) – (iii)
33. Two 40 cm steel wires of different diameters are stretched by equal tensional force of
200 N. The diameter of the first wire is 98% to that of the second wire. When the first
wire is plucked, it produces a sound of 350 Hz. Calculate the
(a) mass per unit length of the first wire. [3 m]
(b) mass per unit length of the second wire. [5 m]
(c) frequency of the sound produced by the second wire when plucked. [2 m]
PSPM 2012/2013 SF016/2 No. 7(c)(ii)
34. A 0.5 kg object is hung at the end of a string of mass 72 g and length 1.2 m. Calculate
the speed of the wave propagation in the string when the system is disturbed. [3 m]
PSPM 2013/2014 SF016/2 No. 7(b)
-1
35. A mechanical wave propagates at 550 m s along a string stretched to a tension of
800 N. The string oscillates at fundamental frequency 440 Hz. Calculate the
(a) mass per unit length of the string. [2 m]
(b) length of the string. [2 m]
(c) frequency of the second overtone and sketch the waveform of the overtone. [3 m]
PSPM 2014/2015 SF016/2 No. 7(b)(i)
36. A teenager would like to double the fundamental frequency of his steel guitar string.
Should he keep to the same string but double the tension or used another string of the
same material and tension but change the diameter of the string? Explain your answer.
[4 m]
PSPM 2016/2017 SF016/2 No. 7(a)(ii)
-1
37. A violin string has a mass per unit length of 0.01 kg m and experiences a tension of
0.36 N. Calculate the wavelength of the string if it vibrates at a frequency of 8 Hz.
[3 m]
PSPM 2017/2018 SF016/2 No. 7(b)
38. A progressive transverse wave travelling on a wire has amplitude of 0.2 mm and a
-1
frequency of 500 Hz with speed of 196 m s .
(a) Write the displacement equation of the wave. [4 m]
-1
(b) If the mass per unit length of the wire is 4.1 g m . Calculate the tension of the
wire. [2 m]
49
PHYSICS PSPM SEM 1 1999 - 2017
Application of Standing Wave ~ Air Column
PSPM JAN 2000/2001 SF035/2 No. 9(c)
39. You are given a closed air column of 80 cm long. When a sound source with a speed of
-1
344 m s is placed near the opened end, stationary wave is generated in the air column.
(a) Draw the stationary wave formed for fundamental tone, first overtone, and second
overtone. [3 m]
(b) Calculate the frequency of fundamental tone and third overtone. [3 m]
PSPM 2002/2003 SF017/2 No. 5
40. Calculate the minimum length of a pipe that has a fundamental frequency of 300 Hz if
the pipe is
(a) closed at one end. [2 m]
(b) opened at both ends. [2 m]
-1
(Speed of sound = 333 m s )
PSPM 2004/2005 SF017/2 No. 5
41. By neglecting the end correction, calculate
(a) the fundamental frequency and the third harmonic frequency for a 0.2 m closed
tube. [2 m]
(b) the second harmonic frequency for a 0.2 m opened tube. [1 m]
-1
(The speed of sound wave in air is 340 m s .)
PSPM 2005/2006 SF017/2 No. 12(b)
42. opened end closed end
S P
x = 0 2 m
m
FIGURE 10.5
FIGURE 10.5 shows a loud speaker S emitting sound wave of frequency 205 Hz in
front of a closed pipe P. The length of the pipe is 2 m and its opened end is at position
x = 0 m. The air column resonates at fifth harmonics.
(a) Draw the stationary wave pattern formed in the air column at its fifth harmonics.
Label all the nodes and antinodes at their respective positions. [3 m]
(b) The maximum amplitude of the stationary wave formed in the pipe is 0.02 m.
Calculate the amplitude for a particle at x = 0.3 m. Neglect end correction factor.
[5 m]
(c) Write an expression in terms of time t for the displacement of the particle in the
stationary wave. [5 m]
50
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