Theory of Machines Unit 1 Objective

MECHANICAL ENGINEERING

1 MECHANISMS, VELOCITYAND ACCELERATION

Class Room Objective

1 The number of degrees of freedom of a planar linkage with 8 links and 9 simple revolute joints is .

(ME-05)

(a) 1 (b) 2 (c) 3 (d) 4

Common data for Q2.Q3, and Q 4

An instantaneous configuration of a four bar mechanism, whose plane is horizontal is shown in the fig

below. At this instant, the angular velocity and angular acceleration of link O2A are = 8 rad/s and = 0,
respectively, and the driving torque () is zero. The link O2Ais balanced so that its center of mass falls at O2

A B
260

100

 60mm 160mm

O2 90o 90o
O4
240mm

2 Which kind of 4 - bar mechanism is O2ABO4 ? (ME-05)
(a) Double crank mechanism (b) Crank-rocker mechanism

(c) Double-rocker mechanism (d) Parallelogram mechanism

3 At the instant considered, what is the magnitude of the angular velocity of O4B ? (ME-05)
(a) 1 rad / s (b) 3 rad / s

(c) 8 rad / s (d) 64 / 3 rad / s

4 At the same instant, if the component of the force at jointAalongAB is 30 N, then the magnitude of the

joint reaction at O2.(ME-05) (b) is 30 N
(a) is zero

(c) is 78 N (d) cannot be determined from the given data

5 In the fig shown, the relative velocity of link 1 with respect to link 2 is 12 m/sec. Link 2 rotates at a

constant speed of 120 rpm. the magnitude of Coriolis component of acceleration of link 1 is.(ME-04)

(a) 302 m/s2. 2
(b) 604 m/s2. 1
(c) 9.6 m/s2.
(d) 1208 m/s2.

2

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THEORY OF MACHINE

6 The fig below shows a planar mechanism with single degree of freedom. (ME-04)

1 2L N

3 4
90o M

1

The instant center 24 for the given configuration is located at a position.

(a) L (b) M (c) N (d) 

7 Match the following:- (ME-04)

Type of Mechanism Motion achieved

P. Scott-Russel mechanism - 1. Intermittent motion

Q. Geneva mechanism - 2. Quick return motion

R. Off-set slider-crank mechanism - 3. Simple harmonic motion

S. ScotchYoke mechanism - 4. Straight line motion

Codes:-

P QRS

(a) 2 3 1 4

(b) 3 2 4 1

(c) 4 1 2 3

(d) 4 3 1 2

8 Match the following with respect to spatial mechanisms. (ME-04)

Type of joint Degrees of Constraint

P - Revolute - 1. Three

Q - Cylindrical - 2. Five

R - Spherical - 3. Four

- 4. Two

- 5. Zero

Codes:

P QR

(a) 1 3 5

(b) 5 4 3

(c) 2 3 1

(d) 4 5 3

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MECHANICAL ENGINEERING

9 In the linkage shown in fig, O1A= 50 mm, AB = 60 mm, O2B = 50 mm, BC = 50 mm, CD = 60 mm,
O3D = 100 mm. In the position shown in the fig, if O1A has momentarily an angular veloctiy of
2 rad/s with out any angular acceleration, then the velocity and acceleration of D will be: (ME-04)

D C O1

(a) 100 mm/s, 100 mm/s2.

(b) 200 mm/s, 100 mm/s2. BA
(c) 200 mm/s, 300 mm/s2. O3 O2
(d) 200 mm/s, 400 mm/s2.

10 Inversions of a mechanism can be obtained by. (PI-03’ )

(a) adding more links (b) changing itslink lengths

(c) fixing itsdifferent links (d) changing link positions/ orientations

Data for Q11 & Q12 are given below. Solve the problems and choose correct answers.
The circular disc shown in its plan view in the fig. rotates in a plane parallel to the horizontal plane about
the point O at a uniform angular velocity . Two other pointsAand B are located on the line OZ at distances
rA and rB from O respectively.

11 The velocity of point B with respect to point Ais a vector of magnitude . (ME-03)
(A) 0
(B)  ( rB- rA) and direction opposite to the direction of motion of point B
(C)  ( rB- rA) and direction same as the direction of motion of point B
(D)  ( rB- rA) and direction being from O to Z.

Z

B

A

O

12 The acceleration of point B with respect to point Ais a vector of magnitude. (ME-03)

(A) 0 rB2- rA2) and direction same as the direction of motion of point B
( rB- rA) and direction opposite to the direction of motion of point
(B) ( ( rB- rA) and direction being from Z to O. B
(C) 2
(D) 2

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THEORY OF MACHINE

13 The minimum number of links in a single degre-of-freedom planar mechanism with both higher and

lower kinematic pairs is. (ME-02)

(a) 2 (b) 3 (c) 4 (d) 5

14 The Coriolis component of acceleration is present in. (ME-02)

(a) 4-bar mechanisms with 4 turning pairs (b) shaper mechanism

(c) slider crank mechanism (d) Scotch Yoke mechanism

15 In a single degree-of-freedom, 4-link, revolute jointed Grashoff mechanism,’l’is the longest link,’s’is

the shortest link and p,q are the two remaining links. These links may be in any order. (ME-02)

(a) write down the condition for feasibility of Grashoff mechanism.

(b) When will you get crank - crank, crank rocker and rocker - rocker type of mechanisms by fixing

links s,l,p or q respectively.

(c) Draw these mechanisms showing motions of crank and rocker links.

16 A four bar mechanism ABCD is shown in the given fig. If the linear velocity ‘V’of the point ‘B’ is

0.5 m/s, then the linear velocity ‘V’ of point ‘C’ will be.

(a) 1.25 m/s E.............0. 1 m
(b) 0.5 m/s 
(c) 0.4 m/s VB 0. 25 m C
(d) 0.2 m/s
B VC D
A =30o

17 Instantaneous center of a body rolling with sliding on a staitonary curved surface lies .
(a) at the point of contact.
(b) on the common normal at the point of contact
(c) on the common tangent at the point of contact
(d) at the center of curvature of the stationary surface

18 The number of degrees of freedom of a five link plane mechanism with five revolute pairs as shown in

the fig. is (93)

(a) 3

(b) 4 4 3
(c) 2 5
(d) 1 2
1

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MECHANICAL ENGINEERING

19 Fig. shows a quick return mechanism. The cranks OA rotates clockwise uniformly. OA 2 cm,
OO1 = 4cm. The ratio of time for forward motion to that for return motion is. (95)

B

(a) 0.5 2 ----------------A
(b) 2.0 O

(c) 12 4
(d) 1 O

20 Arod of length 1 m is sliding in a corner as shown in fig. At an instant when the rod makes an angle of
60o with the horizontal plane, the velocity of point ‘A’on the rod is 1 m/s. The angular velocity of the

rod at this instant is . (96)

(a) 2 rad/s



(b) 1.5 rad/s A 1 m 
(c) 0.5 rad/s
(d) 0.75 rad/s 6 0o
B

21 The number of inversions for a slider crank mechanism is

(a) 6 (b) 5 (c) 4 (d) 3

22 Match the items in columns I and II. (ME-06)

Column I Column II

P. Addendum - 1. Cam

Q. Instantaneous center of velocity - 2. Beam

R. Section modulus - 3. Linkage

S. Prime circle - 4. Gear

Codes:-

P QRS

(a) 4 2 3 1

(b) 4 3 2 1

(c) 3 2 1 4

(d) 3 4 1 2

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THEORY OF MACHINE

23 Match the items in columns I and II (ME-06)

Column I Column II

P. Higher kinematic pair - 1. Grubler’s equation

Q. Lower kinematic pair - 2. Line contact

R. Quick return mechanism - 3. Euler’s equation

S. Mobilityof a linkage - 4. Planer

- 5. Shaper

- 6. Surface contact

Codes:-

P QRS P QRS

(a) 2 6 4 3 (b) 6 2 4 1

(c) 6 2 5 3 (d) 2 6 5 1

24 In a four - bar linkage, S denotes the shortest link length, L is the longest link length, P and Q are the
lengths of other two links.At least one of the three moving links will rotate by 360o if. (ME-06)

(a) S + L  P + Q (b) S + L  P + Q (c) S + P  L + Q (d) S + P > L + Q

25 The input link OP of a four bar linkage is rotated at 2 rad/s counter clockwise direction as shown
below. the angular velocity of the coupler PQ in rad/s, at an instant when O4O2P = 180o , is

(ME =07 )

(a) 4 Q

(b) 2 2 P PQ=O4Q = 2 a
(c) 1
(d) 1 / 2 < O2P=O2Q4 = a

O2 O4

26 If the quick return ratio is 1 : 2 then the length of the crank in mm is . (ME-07)

(a) 250 R Q
(b) 250 3 S
(c) 500  = 2 rad/s
(d) 500  3 O<>

500 mm

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MECHANICAL ENGINEERING

27 The angular speed o PQ in rad/s when the block R attains maximum speed during forward stroke

(stroke with slower speed ) is . (ME-07)

(a) 1/3 (b) 2 / 3 (c) 2 (d) 3

28 A planar mechanism has 8 links and 10 rotary joints. The number of degrees of freedom of the mecha

nism, using Grubler’s criterion, is (ME - 08)

(a) 0 (b) 1 (c) 2 (d) 3

29 Acircular disk of radius ‘R’rolls without slipping at a velocity v. The magnitude of the velocity at point

‘P’ (see fig.) is (ME-08)

3P0o
----------------------------------
--------R----------------- v 

30. For a four - bar linkage in toggle position, the valve of
mechanical advantage is (ME-06)

a) 0.0 b) 0.5 c) 1.0 d) 

31. A simple quick return mechanism is shown in the figure. The forward to return ratio of the quick

return mechanism is 2 : 1. If the radius of the crank O1 P is 125 mm, then the distance ‘d’
(in mm) between the crank centre to lever pivot centre point should be (GATE-ME-09)

a) 144.3

b) 216.5

c) 240.0

d) 250.0

32. Mobility of a statically indeterminate structure is (GATE-ME-10)
a) < - 1
b) 0
c) 1
d) > 2

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THEORY OF MACHINE

33. There are two points P and Q on a planar rigid body. The relative velocity between the two

points (GATE-ME-10)

a) should always be along PQ

b) can be oriented along any direction

c) should always be perpendicular to PQ

d) should be along QP when the body undergoes pure translation

34. Which of the following statements is INCORRECT (GATE-ME-10)

a) Grashof’s rule states that for a planar crank-rocker four bar mechanism, the sum of the

shortest and longest link lengths cannot be less than the sum of the remaining two link lengths.

b) Inversion of a mechanism are created by fixing different links one at a time

c) Geneva mechanism is an intermittent motion device

d) Gruebler’s criterion assumes mobility of a planner mechanism to be one.

35. A double – parallelogram mechanism is shown in the figure. Note that PQ is a single link. The

mobility of the mechanism is

a) -1

b) 0

c) 1

d) 2

36. A slide disc of radius r rolls without slipping on a horizontal floor with angular velocity  and

angular acceleration  . The magnitude of the acceleration of the point of contact on the disc
is (GATE-ME-12)
a) zero b) r

c) r 2  r 2 d) r 2

37. In the mechanism given below , if the angular velocity of the eccentric circular

disc is 1 rad/s, the angular velocity (rad/s) of the follower link for the instant shown in the

figure is (GATE-ME-12)

Note : All dimensions are in mm

a) 0.05

b) 0.1

c) 5.0

d ) 10.0

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MECHANICAL ENGINEERING

38. A link OB is rotating with a constant angular velocity of 2 rad/s in counter clockwise direction

and a block is sliding radially outward on it with an uniform velocity of 0.75 m/s with respect to

the rod, as shown in the figure below. IF OA = 1 m, the magnitude of the absolute acceleration

of the block at location A in m/s2 is (GATE-ME-13)

a) 3

b) 4

c) 5

d) 6

39. A planner closed kinematic chain is formed with rigid links PQ = 2.0 m, QR = 3.0 m, RS = 2.5 m

and SP = 2.7 m with all revolute joints. The link to the fixed to obtain a double rocker (rocker-

rocket) mechanism is (GATE-ME-13)

a) PQ b) QR c) RS d) SP

40. A circular object of radius r rolls without V slipping on a horizontal level floor with the center

having velocity V. The velocity at the point of contact between the object and the floor is

(GATE-ME-14)

a) zero

b) V in the direction of motion

c) V opposite to the direction of motion

d) V vertically upward from the floor

41. A rigid link PQ is 2 m long and oriented at 200 to the horizontal as shown in the figure. The

magnitude and direction of velocity VQ, and the direction of velocity VP are given. The magnitude
of VP (in m/s) at this instant is (GATE-ME-14)

a) 2.14

b) 1.89

c) 1.21

d) 0.96

42. A 4-bar mechanism with all revolute pairs has link lengths lf = 20 mm, lin = 40 mm, lco = 50 mm
and lout = 60 mm. suffixes ‘f’, ‘in’, ‘co’ and ‘out’ denote the fixed link, the input link, the coupler
and output link respectively. Which of the following statements is true about the input and

output links? (GATE-ME-14)

a) Both links can execute full circular motion

b) Both links cannot execute full circular motion

c) Only the output link cannot execute full circular motion

d) Only the input link cannot execute full circular motion

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THEORY OF MACHINE

43. For the given statements :
I. Mating spur gear teeth is an example of higher pair
II. A revolute joint is an example of lower pair

Indicate the correct answer (GATE-ME-14)

a) Both I and II are false

b) I is true and II if false

c) I is false and II is true

d) Both I and II are true

44. A wheel of radius r rolls without slipping on a horizontal surface shown below. If the velocity of

point P is 10 m/s in the horizontal direction, the magnitude of velocity of point Q (in m/s) is

_______ (GATE-15-Set-1)

45. In the figure, link 2 rotates with constant angular velocity 2 . A slider link 3 moves outwards
with a constant relative velocity VQ/P, where Q is a point on slider 3 and p is a point on link 2.
The magnitude and direction of Coriolis component of acceleration is given by
(GATE-15-Set 3)

a) 2 2 VQ/P; direction of VQ/P, rotated by 900 in the direction of
b) VQ/P; direction of VQ/P, rotated by 900 in the direction of
c) 2 VQ/P; direction of VQ/P, rotated by 900 Opposite to the direction of
d) VQ/P; direction of VQ/P, rotated by 900 Opposite to the direction of

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46. A rigid link PQ is undergoing plane motion as shown in the figure (VP and VQ are non-zero). VQP

is the relative velocity of points Q with respect to point P. (GATE-16-Set 3)

Which one of the following is TR?UE ?

a) VQP has components along and perpendicular PQ
b) VQP has only one component directed from P to Q
c) VQP has only one component directed from Q to P
d) VQP has only one component perpendicular to PQ

47. The number of degrees of freedom in a planner mechanism having a links and j simple hinge

joints is (GATE-16-SET-3)

a) 3 (n -3) – 2j b) 3 (n – 1) – 2j c) 3n – 2j d) 2j – 3n + 4

48. Match the approaches given below to perform stated kinematics /dynamics analysis of machine

(GATE-ME-09)

Analysis Approach

P. Continuous relative rotation 1. D’ Alembert’s principle

Q. Velocity and acceleration 2. Grubler’s criterion

R. Mobility 3. Grashoff’s law

S. Dynamic – static analysis 4. Kennedy’s theorem

. PQRS
a) 1 2 3 4
b) 3 4 2 1
c) 2 3 4 1
d) 4 2 1 3

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THEORY OF MACHINE

49. For the configuration shown, the angular velocity of link AB is 10 rad/s counterclockwise. The
magnitude of the relative sliding velocity (in ms-1) of slider B with respect to rigid link CD is
(GATE-ME-10)
a) 0

b) 0.86

c) 1.25

d) 2.50

50. For the four – bar linkage shown in the figure, the angular velocity of link AB is 1 rad/s. The

length of link CD is 1.5 times the length of link AB. In the configuration shown, the angular

velocity of link CD in rad/s is (GATE-ME-11)

a) 3 b) 3
2

c) 1 d) 2
3

51. A slider-crank mechanism with crank radius 60 mm and connecting rod 240 mm is shown in the
figure. The crank is rotating with a uniform angular speed of 10 rad/s , counter clockwise. For
the given configuration, the speed (in m/s) of the slider is ___________ . (GATE-ME-14)

52. An offset slider – crank mechanism is shown in the figure at an instant. Conventionally, the

Quick Return Ratio (QRR) is considered to be greater than one. The value of QRR is

______________ (GATE-ME-14)

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MECHANICAL ENGINEERING

53. In a certain slider crank mechanism, lengths of crank and connecting road are equal. If the

crank rotates with a uniform angular speed of 14 rad/s and the crank length is 300 mm, the

maximum acceleration of the slider (in m/s2) is ____________. (GATE-15-Set 2)

54. The number of degree of freedom of the linkages shown in the figure is (GATE-15-Set 3)

a) -3
b) 0
c) 1
d) 2
55. Figure shows a wheel rotating, about O2, Two points A and B located along the radius of wheel
have speed s of 80 m/s and 140 m/s respectively. The distance between the points A and B is
300 mm. The diameter of the wheel (in mm) is ____________________ (GATE-15-Set 3)

56. The an off-set slider crank mechanism, shown in figure, the crank is rotated at a constant

speed of 150 rpm. The value of the angle  shown in the figure is 200. What is the ratio of

forward to return stroke time ? Can this mechanism be used in an application involving quick

return ? (GATE-PI-15)

a) 3.33, No
b) 0.73, yes
c) 1.25, yes
d) 0.73, No

57. The rod AB, of length 1 m shown in the figure is connected to two slider to two slider at each
end through pins. The slider can slide along QP and QR. If the velocity VA of the slider at A is 2
m/s, the velocity of the midpoint of the rod at this instant is __________ m/s. (GATE-16- Set 2)

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