EP015 TUTORIAL PART 2

TUTORIAL 6 - TUTORIAL 10

6 CIRCULAR MOTION
7 GRAVITATION
8 ROTATION OF RIGID BODY
9 SIMPLE HARMONIC MOTION
10 MECHANICAL AND SOUND

WAVES

EP015 PHYSICS 1 25

TUTORIAL 6: CIRCULAR MOTION
LEARNING OUTCOMES

UNIFORM CIRCULAR MOTION
1. Describe uniform circular motion.
2. Convert units between degrees, radian, and revolution or rotation.

CENTRIPETAL FORCE
3. Define centripetal acceleration.
4. Solve problems related to centripetal force for uniform circular motion cases:

horizontal circular motion, vertical circular motion and conical pendulum.

EP015 PHYSICS 1 26

TUTORIAL 6: CIRCULAR MOTION

OBJECTIVE QUESTIONS
1. Which of these statements is incorrect about uniform circular motion?
A. The speed is always constant
B. The velocity is changing with time
C. The object is speeding up or slowing down
D. The direction of the velocity is changing with time

2. The blades of a table fan make 25 revolutions in one minute. What is the
angular velocity of the blades?

A. 1.31 rad s-1
B. 2.62 rad s-1
C. 78.5 rad s-1
D. 157 rad s-1

3. The three hands on a clock are called the hour hand, the minute hand and the
second hand. What is the angular velocity of the second hand?

A. 0.105 rad s-1
B. 1.05 rad s-1
C. 5.1 rad s-1
D. 1.5 rad s-1

4. A mass moves in a horizontal circle with a constant speed of 1. When the
same mass moves in the same circular path with a speed of 2 , the new
centripetal force is half that of the first. What is the value of 2 ?

1

A. 0.5
B. 0.7
C. 1.4
D. 2.0

SUBJECTIVE QUESTIONS
1.
a. With the help of a diagram, describe uniform circular motion
b. Define centripetal acceleration
c. What is meant by centripetal force

2. A gramophone rotates at 33 1 rev min-1 and has a radius of 0.15 m. What is:

3

a. its angular velocity
b. the speed of a point on its circumference
c. the centripetal acceleration of a point on its circumference

3. Calculate the centripetal acceleration of an object at the Equator due to
rotation of the Earth. The radius of Earth is 6.37 x 106 m.

EP015 PHYSICS 1 27

4. An object of mass 0.20 kg on the end of a string is swung around in a
horizontal circle of radius 0.80 m with a constant speed of 4.0 m s-1. What is
the tension in the string?

5. What is the speed of an aircraft moving in a horizontal circle of radius 300 m if
the acceleration experienced by the pilot is 4g?

6. Name the force that is responsible for the following motion:
a. A satellite orbiting the Earth
b. A ball attached to a string that swirls horizontally

7. An object of 2.0kg mass is whirled in a vertical circle with a constant speed of
6.0 m s-1. Given the radius of the circle is 2.5m
a. Determine the centripetal force of the object
b. The maximum and minimum tension of the string circle.

8. Figure below shows a child riding on a Ferris Wheel “Eye on Malacca” of 4.5m
in diameter, rotating at constant speed.
a. Draw the free body diagram and write equations for forces acting on
the passenger at position A and C
b. What is the maximum angular speed so that the passenger is not
thrown out from the chair?

9. A car travels at a constant speed of 13.4 m s-1
on a level circular turn of radius 50.0m, as
shown in the bird’s-eye view in Figure below.
What minimum coefficient of static friction, µs,
between the tires and roadway will allow the
car to make the circular turn without sliding?

EP015 PHYSICS 1 28

10. A student is to swing a bucket of water in a vertical circle without spilling any.
If the distance from his shoulder to the center of mass of the bucket of water
is 1.0 m, what is the minimum speed required to keep the water from coming
out of the bucket at the top of the swing?

11. A ball of mass m moves with constant velocity v, in a horizontal circular path
of radius r. The ball is attached to a string of length l=24 cm and makes a
conical pendulum as shown in figure below.

a. Sketch a Free Body Diagram showing the forces acting on the object.
b. If θ=30⁰, calculate the

i. radius
ii. velocity
12. A model airplane of mass 0.750 kg flies with a speed of 35.0 m s-1 in a
horizontal circle at the end of a 60.0 m control wire as shown in Figure (a).
The forces exerted on the airplane are shown in Figure(b); the tension in the
control wire, θ=20.0° inward from the vertical. Compute the tension in the
wire, assuming the wire makes a constant angle of θ=20.0° with the
horizontal.

EP015 PHYSICS 1 29

SUGGESTED ANSWER

OBJECTIVE: C, B, A, B
SUBJECTIVE QUESTIONS:
2. 3.49 rad s-1; 0.52 m s-1; 1.83 m s-2
3. 3.37 x 10-2 m s-2
4. 4 N
5. 108.5 m s-1
7. 28.8N; 48.42N, 9.18N
8. 2.09rad/s
9. µs=0.366
10. 3.13 m s-1
11. 0.12 m; 0.824 m s-1
12. 12.8 N

EP015 PHYSICS 1 30

TUTORIAL 7: GRAVITATION
LEARNING OUTCOMES

GRAVITATIONAL FORCE AND FIELD STRENGTH

1. State and use the Newton's law of gravitation F  G Mm
r2

2. Define and use gravitational field strength ag GM
r2

GRAVITATIONAL POTENTIAL ENERGY

3. Define gravitational potential energy.
4. Use gravitational potential energy U   GMm

r

SATELLITE MOTION IN A CIRCULAR ORBIT

5. Use escape velocity vesc  2GM  2gR
R

6. Use equation for satellite motion

 velocity v  GM
r

 period, T  2 r 3
GM

EP015 PHYSICS 1 31

TUTORIAL 7: GRAVITATION

Universal gravitation constant, G = 6.67 x 10-11 N.m2.kg -2
Mass of earth = 6.0 x 1024 kg
Radius of earth = 6.4 x 103 km

1. Two objects with masses m1 and m2 respectively are a distance r apart. The
magnitude of the gravitational force between them is F. The masses are
changed to 2m1 and 2m2 and the distance is changed to 4r. Find the magnitude
of the new gravitational force in terms of F?

A. C.

B. D.

2. A satellite is in a circular orbit about the Earth at an altitude at which air
resistance is negligible. Which of the following statements is TRUE?

A There is only one force acting on the satellite.
B. There are two forces acting on the satellite, and their resultant is zero.
C There are two forces acting on the satellite, and their resultant is not zero.
D There are three forces acting on the satellite.

3. A spaceship of mass m circles a planet of mass M in an orbit of radius R. How
much energy is required to transfer the spaceship to a circular orbit of radius
3R?

A C.

B D.

4. a) i) State Newton’s law of gravitation.
b) ii) Define gravitational field strength
Sketch a graph showing the variation of acceleration due to gravity g
with distance r from the center of the earth.

5. A 200-kg object and a 500-kg object are separated by 0.400 m.

a) Determine the net gravitational force exerted on a 50.0 kg object

placed midway between them. [2.5 x 10-5 N]

b) At what position does the 50.0 kg object be placed so as to experience

a net force of zero? [0.155 m]

6. a) What is meant by escape velocity?
b) Calculate the kinetic energy needed by a space ship of mass 6×103 kg
to escape from the gravitational force of the Earth. [3.74 x 1011 J]

EP015 PHYSICS 1 32

7. A satellite of mass 66 kg is moving in an orbit at a distance of 5.7R above the
Earth’s surface, where R is the value of the mean radius of the Earth. If the
gravitational field strength at the Earth’s surface is 9.81 N kg-1, calculate

a) the centripetal force acting on the satellite. [14.4 N]

b) the period of the satellite in orbit (in hours). [24.5 hours]

8. Find the reading on a spring scale for the weight of a 56 kg women in a lift that

moves

a) with constant speed of 3.0 m s-1, upward [549 N]

b) with constant speed of 3.0 m s-1, downward [549 N]

c) with acceleration of 0.33g, upward [731 N]

d) with acceleration of 0.33g downward and [368 N]

e) in free fall. [0]

9. A communication satellite of mass 50 kg is launched so that its orbit is directly

above the equator. Its angular velocity is the same as that of Earth. As a result,

the satellite is always directly above a point on Earth. Determine

a) the angular velocity ( in radian per second) of the rotation of the Earth

about its axis. [7.27 x 10-5 rad/s]

b) the radius of the orbit. [1.04 x 1011 m]

10. The average radius of the orbit of Earth round the sun is 1.5×1011 m. The orbit

can be considered to be circular. Calculate

a) the mass of the sun, [2.0 x 1030 kg]

b) the gravitational potential energy of a helium atom on the surface of the Sun,

[- 1.19 x 10-15 J]

c) the minimum velocity for a helium atom to escape from the gravitational field

of the sun. [6.0 x 105 m s-1]

[Given: radius of the sun = 7.0 x 108 m; mass of helium atom = 6.6×10-27 kg]

EP015 PHYSICS 1 33

TUTORIAL 8: ROTATION OF RIGID BODY
LEARNING OUTCOMES

ROTATIONAL KINEMATICS

1. Define and use:

 angular displacement,

 average angular velocity, av

 instantaneous angular velocity,

 average angular acceleration,  av

 instantaneous angular acceleration, 

2. Relate parameters in rotational motion with their corresponding quantities in

linear motion:

s  r ; v  r ; at  r ; ac  r 2  v2
r

3. Solve problems related to rotational motion with constant angular acceleration

  0  t ;   0t  1 t 2 ; 2  02  2
2

EQUILIBRIUM OF A UNIFORM RIGID BODY

4. Define torque   r  F

5. Solve problems related to equilibrium of a uniform rigid body.

ROTATIONAL DYNAMICS

6. Define and use the moment of inertia of a uniform rigid body.
7. State and use torque   I

CONSERVATION OF ANGULAR MOMENTUM

8. Define and use angular momentum L  I
9. State and use principle of conservation of angular momentum.

EP015 PHYSICS 1 34

TUTORIAL 8: ROTATION OF RIGID BODY
1. Kinematics :

A wheel starts to rotate from rest and attains an angular speed 10.0 rpm in
15.0 s. By using a suitable figure, determine :

a) the angular acceleration of the wheel.
b) the angular displacement of the wheel.

2. Moment of inertia of particles :
Determine the moment of inertia of the system of particles (connected by a
light rod) as shown in FIGURE 1 about an axis at :

FIGURE 1

a) point A.
b) point B.
c) point C.

3. Torque :
A disc of mass 2.0 kg and radius 10.0 cm is pivoted at its centre of mass. Two
external forces act on the disc as shown in FIGURE 2 below.

35o F1 = 15 N F2 = 5 N
30o

•

axis of rotation

FIGURE 2

Determine the total torque produced due to the forces.
Determine the angular acceleration of the disc.
If the disc is initially at rest, determine the number of complete revolution after
1 minute.

4. Equilibrium of Rigid Body :
FIGURE 3 shows a light rod of length 1.5 m which is pivoted at its centre. A
force of 12 N is exerted at one end of the rod at such direction as shown in
the figure.

F = 12N 200

FIGURE 3

EP015 PHYSICS 1 35

a) Determine the magnitude and direction of the torque produced by the
force.

b) Determine the direction of rotation due to the force.
c) If you are given a second force of magnitude 10 N, suggest any

position and determine the direction of the force (draw the force on the
figure) to balance the system.

5. Equilibrium of Rigid Body :
A uniform beam of length 5.0 m and has mass 12 kg is supported by two
pivots as shown in FIGURE 4.

R1 R2
3m

FIGURE 4

a) Show all forces acting on the beam.
b) Determine the magnitude of reaction force, R1.
c) Determine the magnitude of reaction force, R2.

6. Equilibrium of Rigid Body :
FIGURE 5 shows a uniform 50 kg beam of length 1.5 m. The left and of the
beam is hinged to a rough wall and is supported at the other end by a light
cable.

20o

1.2 m
W

FIGURE 5

a) Show all forces acting on the beam.
b) If the cable can support 1500 N, what is the maximum mass of W?

7. Equilibrium of Rigid Body :
A ladder of length 2.5 m and of mass 12 kg leans at an angle 20o against a
smooth wall. The lower end of the ladder in on the rough floor.
a) Show all forces acting on the ladder.
b) Determine the force on the upper end of the ladder.
c) Determine the resultant force on the lower end of the ladder.

EP015 PHYSICS 1 36

8. Moment of inertia of a body:
a) A solid sphere of mass 0.2 kg and radius 10.0 cm is connected to a
solid cylinder of mass 0.3 kg and radius 1.0 cm through a light rod of
length 50.0 cm. By using a suitable figure, determine the moment of
inertia of the system about an axis at the centre of the rod.
b) A disc has mass of 2.0 kg and radius of 10.0 cm. By using a suitable
figure, determine the moment of inertia of the disc about an axis 0.2 cm
from its centre of mass.

9. Conservation of Angular Momentum
A merry-go-round of mass 150 kg and diameter 3.0 m is initially rotating
uniformly 10rpm. Then, a boy of mass 25.0 kg jumps onto the edge of the
‘disc’. By using a suitable figure, determine:
a) the moment of inertia of the boy before and after jumping on the disc.
b) The new angular velocity of the system.

Suggested Answers

1. (a) α = 0.069 rad s-2 (b) θ = 7.85 rad

2. (a) IA = 0.59 kg m2 (b) IB = 0.19 kg m2 (c) IC = 0.34 kg m2

3. (a)  T = 1.11 Nm (b) α = 111 rad s-2 (c) N = 31,799 rev
4. (a)  = 8.46 Nm

5. (a) (b) R1 = 19.62 N (c) R2 = 98.10 N

6. (a) (b) W = 34.12 kg

7. (a) (b) R = 21.42 N (c) R = 119.65 N

8. (a) I = 0.031 kg m2 (b) I = 0.01 kg m2

9. (a) If = 56.25 kg m2 (b) ω = 0.79 rad s-1

EP015 PHYSICS 1 37

TUTORIAL 9: SIMPLE HARMONIC MOTION
LEARNING OUTCOMES

KINEMATICS OF SIMPLE HARMONIC MOTION
1. Explain SHM.
2. Solve problems related to SHM displacement equation, y  Asin t
3. Apply equations:

 velocity v  dy   A2  y 2
dt

 acceleration a  dv  d 2 y   2 y
dt dt 2

 kinetic energy K  1 m 2 (A2  y 2 )
2

 potential energy U  1 m 2 y 2
2

4. Emphasise the relationship between total SHM energy and amplitude.
5. Apply velocity, acceleration, kinetic energy, and potential energy for SHM.

GRAPHS OF SIMPLE HARMONIC MOTION
6. Discuss the following graphs

 displacement-time
 velocity-time
 acceleration-time
 energy-displacement
PERIOD OF SIMPLE HARMONIC MOTION
7. Use expression for period of SHM, T for simple pendulum and spring.

EP015 PHYSICS 1 38

TUTORIAL 9: SIMPLE HARMONIC MOTION

1. The amplitude of a system moving with SHM is doubled. The total energy E will

then be

A. E C. 3E

B. 2E D. 4E

2. A weight of mass m is at rest at O when suspended from a spring as shown in
FIGURE 1. When it is pulled down and released, it oscillates between positions A
and B. Which of the following statements about the system consisting of the spring
and the mass is CORRECT?

FIGURE 1

A. The gravitational potential energy of the system is greatest at A.
B. The elastic potential energy of the system is greatest at O.
C. The rate of change of momentum has its greatest magnitude at A and B.
D. The rate of change of gravitational potential energy is smallest at O.

3. A graph of position versus time for an object oscillating at the free end of a
horizontal spring is shown in FIGURE 2. The point at which the object has negative
velocity and zero acceleration is

Q
PR

S

A. P FIGURE 2
B. Q C. R
D. S

EP015 PHYSICS 1 39

4. A ball moves in a circular path of diameter 0.15 m with constant angular
speed of 20 rpm. Its shadow performs SHM on the wall behind it. Calculate the
acceleration and speed of the shadow at

a) the turning point of the motion,
b) the equilibrium position and
c) a point 6 cm from the equilibrium position.

5. A piston in a gasoline engine undergoes SHM. The extremes of its position
relative to its equilibrium point are 5.00 cm and -5.00 cm. The engine is running at
the rate of 3600 rpm. Calculate,

a) maximum velocity
b) maximum acceleration.

6. A block of unknown mass is attached to a spring with a spring constant of
6.50 N m-1 and undergoes SHM with amplitude of 10.0 cm. When the block is
halfway between its equilibrium position and the endpoint, its speed is measured to
be 30.0 cm s-1. Calculate,

a) mass of the block,
b) period of the motion and
c) maximum acceleration.

7. The position of a particle is given by the expression, x = (4.00 m) cos (3.00 t)

where x in meters and t in seconds. Calculate

a) frequency
b) period
c) amplitude
d) the position of the particle at t=0.25 s

8. A 50.0-g object connected to a spring with a force constant of 35.0 N m-1
undergoes SHM on a horizontal, frictionless surface with amplitude of 4.00 cm.
Calculate,

a) total energy of the system ,
b) speed of the object at x = 1.00 cm ,
c) kinetic energy at x = 3.00 cm and
d) potential energy x = 3.00 cm.

EP015 PHYSICS 1 40

9. A particle moving along the x axis in SHM starts from the origin at t = 0 and

moves to the right. The amplitude of its motion is 2.00 cm and the frequency is

1.50 Hz. Show that the position of the particle is given by x = (0.02) sin(3.00 t)
a)

where x is in meters and t is in seconds.

Calculate, maximum speed and the earliest time (t > 0) at which the particle has
b) this speed
maximum acceleration and the earliest time (t > 0) at which the particle
c) has this acceleration and
total distance traveled between t = 0 and t = 1.00 s.
d)

10. a) Sketch a graph showing the variation of energies against displacement
b) x for a system undergoing SHM.
A particle of mass 4 kg is vibrating in SHM. The graph for the potential
energy U against displacement x is shown in FIGURE 3.

U(J)
1.0

-0.2 0 0.2 x (m)
FIGURE 3
Calculate :
i) angular velocity and
ii) period.

EP015 PHYSICS 1 41

ADDITIONAL PROBLEMS

1. A particle perfoms simple harmonic motion with amplitude 2.0 103m and period

0.10 s. The maximum speed of the particle is

A. 0.08ms1 C. 0.21ms1

B. 0.13ms1 D. 0.35ms1

2. If the length of a simple pendulum is 3 times its original length, the ratio of the
new frequency to the original frequency is
11

A. 3 C. 3
1

B. 9 D. 3

3. When a particle performs simple harmonic motion, the velocity and displacement

shows a phase difference of

 3 rad
rad C. 4

A. 4

 D. rad
rad

B. 2

4. A particle moves with simple harmonic motion along a staright line with
amplitude of 0.03m and period of 6 seconds. Calculate
a) the maximum velocity
b) the maximum acceleration of the particle

5. When the load of a spring displaced vertically 3.0 cm from its equilibrium
position and then released, the load is seen oscillating in simple harmonic motion. If
the period of oscillation is 2.0s, find the distance moved in

a) the first 1.0s
b) the first 0.75s

6. A particle is perform simple harmonic motion with its displacement x from the
equilibrium position according to the relationship of x  Asin t . Sketch a graph to
show the relationship between

a) velocity and time
b) kinetic energy and time

7. A body moves in SHM with an amplitude of 30 mm and a frequency of 2.0 Hz.
Calculate the values of

i) acceleration at the center and extremities of the oscillation.
ii) Velocity at these positions
iii) Velocity and acceleration at a point midway between the center and

extremity of the oscillation.

EP015 PHYSICS 1 42

SUGGESTED ANSWER

TUTORIAL

4. a) 0 ms-1 , 0.329 ms-2 b) 0.157 ms-1 , 0 ms-2
c)
9.42 x 10-2 ms-1 , 0.263 ms-2
5. a)
6. a) 18.8 m s-1 b) 7.11 x 103 m s-2
7. a)
0.542 kg b) 1.81 s c) 1.20 m s-1
c) d) 12.0 cm
8. a) 1.50 Hz b) 0.667 s
4.00 m d) −2.83 m
c)
2.80x10-2 J b) 1.02 m s-1
9.
10. b) 1.22x102 J d) 1.58x10-2 J

b) t1 t 1
18.8 cm s-1 , 3 s
c) 178 cm s-2, 2 s

i. 3.54 rad s-1 ii. 1.78 s

ADDITIONAL PROBLEMS

1. B b) 5.1 cm
2. A
3. B
4. a) 6.0 cm
6. 1.10 s

8 c) T  2 i d)   10 2rads1
9. i) c

iii) a  0.48 2ms2 ii) v = 0
10. b)
v  0.33ms1 ; a  0.24 2ms2

f  2.2Hz

EP015 PHYSICS 1 43

TUTORIAL 10: MECHANICAL AND SOUND WAVES
LEARNING OUTCOMES

PROPERTIES OF WAVES

1. Define wavelength and wavenumber.
2. Solve problems related to equation of progressive wave, y(x, t)  Asin(t  kx)
3. Discuss and use particle vibrational velocity and wave propagation velocity.
4. Discuss the graphs of:

 displacement-time, y-t
 displacement-distance, y-x

SUPERPOSITION OF WAVES

5. State the principle of superposition of waves for the constructive and destructive
interferences.

6. Use the standing wave equation, y  Acos kxsin t
7. Discuss progressive and standing wave.

SOUND INTENSITY

8. Define and use sound intensity.
9. Discuss the dependence of intensity on amplitude and distance from a point

source by using graphical illustrations.

APPLICATION OF STANDING WAVES

10. Solve problems related to the fundamental and overtone frequencies for:
 stretched string
 air columns (open and closed end)

11. Use wave speed in a stretched string, v  T


DOPPLER EFFECT
12. State Doppler Effect for sound waves.

13. Apply Doppler Effect equation f0   v  v0  for relative motion between source
v  vs

and observer.

EP015 PHYSICS 1 44

TUTORIAL 10: MECHANICAL AND SOUND WAVES

Answer all the questions.

1. A progressive transverse wave is given as y = 20 sin (6x – 6t) where
displacement, y in centimetres, distance along the wave, x is in metres and
time, t is in seconds. At x = 2 m and t = 1 s, determine
a. the displacement and
b. the velocity of the wave.

2. A sinusoidal wave is described by
y =20 sin (0.5x -30t)

where y and x is in millimeters and t is in second. Determine its
i. Amplitude
ii. angular frequency
iii. wave number
iv. wave velocity
v. direction of motion

3. The equation of a progressive wave is given by y = 6 sin(5πt - 0.8x). Write the
equation for particle velocity and determine the maximum velocity of the
particle.

4. y1 = 0.075 sin (4πt - πx )
y2 = 0.075 sin (4πt + πx)

Both progressive waves above form a standing wave where y1 and y2 is in
meters and t is in seconds

a. Write the expression representing the new wave.
b. Calculate the maximum velocity of wave.

5. a. Sketch a graph to represents the variation of the intensity,I with distance r
from a source that emits sound uniformly in all directions.
b. The intensity of sound from a speaker at a distance of 150 cm to the left of
the speaker is 180 Wm2. Determine the sound energy produced by the
speaker for every 1 second.

6. Assume that the sound from a firework that explodes in air spreads out
uniformly
in all directions. If the sound reaches a listener 600 m away has an intensity of
10 x 10-8 W m-2
i. What happen to the intensity of the sound if the distance between the
explosion
and the listener is quadrupled while the power of the explosion remains
constant?
ii. What is the intensity detected by another listener at a distance 150 m away
from the explosion.

EP015 PHYSICS 1 45

7. One of the strings on a guitar has a linear density of 5.28 x 10-3 kg m-1 and is
stretched with a tension of 226 N. The string produces the muscial note E
when vibrating its entire length at the fundamental frequency of 164.8 Hz.
a. What is the length of the string?
b. Where the player must presses his finger so as to produce a note of
fundamental frequency 329.6 Hz?

8. An organ pipe has a length of 0.75 m. (Speed of sound in air is 343 m s-1)
a. What would be the length of a closed organ pipe whose third harmonic
is the same as the fundamental frequency of the open pipe?
b. Sketch the profile of second overtone of the open organ pipe.

9. A pipe that is open at both ends has a fundamental frequency of 300 Hz when
the speed of sound in air is 333 ms-1.
a. What is the length of the pipe?
b. What is the frequency of second harmonic when the temperatuere of
the air is increased so that the speed of sound in the pipe is 344 m s-1?

10. The frequency of an ambulance siren is 700 Hz. What are the frequencies
heard
by a stationary pedestrian as the ambulance,(Assume that vsound =343 m s-
1)
a. the ambulance approaches her at a speed of 90 km h-1
b. the ambulance passes her at a speed of 90 km h-1.

Suggested Answers:

1. 0 m, 1 ms-1
2. 20 mm, 30 rads-1, 0.5 mm-1, 60 mms-1, to the right
3. 30π cos(5πt - 0.8x) ms-1, 30π ms-1
4. Y = 0.15 sin(4πt) cos(πx), 4 ms-1
5. 5089 W
6. 1.60 x 10-6 Wm-2
7. 0.628 m, 0.314
8. 1.13 m
9. 0.56 m, 614.3 Hz
10. 652.4 Hz, 755 Hz