Kinematics of rigid bodies

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P
INTRODUCTION
In this Chapter we will do motion analysis of rigid bodies without involving
the forces responsible for the motion. We will learn to find the position,
velocity and acceleration of the different particles which together form a rigid
body.
In kinematics of particles the size or dimensions of the body were not taken
into consideration during motion analysis, since we treated the entire body
irrespective of its size (as large as a car, lift, train or airplane) as one single
particle. In rigid body kinematics, we shall involve the size and dimensions of
the body also. Therefore, now the body would be considered as made up of
several particles, connected to each other, such that their relative positions
do not change, as the body performs its motion.
There are various types of rigid body motion. We will begin with first
classifying them and then we will study each type in detail and thereby
develop our base for study of kinetics of rigid bodies.
Types of Rigid Body Motion
The motion of the rigid body can be classified under the following types,
1) Translation
2) Rotation about fixed axis
3) General plane motion
4) Motion about a fixed point
5) General motion
TRANSLATION MOTION
In this type of motion all the particles forming the body travel along parallel
paths. Also the orientation of the body does not
change during motion. Translation motion is
further classified as:
a) Rectilinear Translation
b) Curvilinear Translation
Kinematics of Rigid Bodies

2
Rectilinear Translation
In this type of translation motion, all the particles travel along straight
parallel paths. Consider the motion of a body from position (1) to position (2).
Here we note two things, firstly a straight line joining particles P and Q in
position (1) maintains the same direction in position (2) This indicates that
the orientation of the body remains the same throughout the motion. The
second thing which is special to rectilinear translation motion, is that the
path described by the various particles, such as P and Q in our example, are
parallel and they are straight.
Examples of rectilinear translation motion are,
1) A piston which travels in the fixed slot.
2) Motion of train on a straight track.
3) Motion of lift etc.
Curvilinear Translation
In this type of translation motion, all the particles travel along curved parallel
paths. Consider the motion of a body from position (1) to position (2)
Here we note that the line joining particles P and Q in position (1) has the
same orientation in position (2) This is indicative of translation motion of the
body. Further the path described by all the particles, such as P and Q in our
example, are parallel and they are curved. This indicates curvilinear
translation motion.
At a given instant, for a translation motion, both rectilinear and curvilinear,
all the particles of the body have same velocity and same acceleration.
P Qi.e. v = v …… [13.1 (a)]
P Qand a = a …… [13.1 (b)]

3
ROTATION ABOUT FIXED AXIS
In this motion all the particles forming the body
travel along circular paths of different radii,
around a common centre. This common point is
known as centre of rotation. An axis
perpendicular to the plane of the body and
passing through the centre of rotation is known
as the axis of rotation. Since the axis of rotation
remains stationary or fixed, we call such motion
as Rotation about Fixed Axis.
Important Terms
1) Angular Position
Let O, be the centre of rotation and z axis be
the fixed axis of rotation. Let the x axis be the
reference axis with respect to which we
measure the position of the rotating body. Let
us mark an arbitrary point P on the body.
The angle  measured in the anticlockwise
direction which the line OP makes with the x
axis would then be the angular position of the
particle.
As the body rotates and occupies position (2), the angular position of the
body also changes and would have a new value of . Angular position is
measured in units of radians (rad). It may also be measured in units of
revolution or degree. These are related as
1 revolution = 2 rad = 360
For example, if the body’s initial angular position is 30 0.5236 rad and it
completes two revolutions, its new angular position would be,
750 13.09 rad .

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2) Angular Displacement
The change in angular position of the body during its motion is known as
the angular displacement of the body.
If 1 , is the angular position of the body in position (1) and if this changes
to 2 at position (2), the angular displacement of the body is,
2 1=   
3) Angular Velocity
The rate of change of angular position with respect to time is the angular
velocity of the rotating body.
d
dt

  ……. [13.2]
The magnitude of angular velocity is denoted by notation  (omega) and its
direction acts along the axis of rotation, the sense being defined by right
hand rule. i.e. for a body in the x-y plane rotating about a fixed axis
parallel to the z axis in the anticlockwise direction, will have positive
angular velocity. The same body, if its rotates in the clockwise direction
will have negative angular velocity. Angular velocity is represented by
curved arrow representing clockwise or anticlockwise sense.
Units of angular velocity are rad/s, through other unit like revolutions per
minute (rpm) is also commonly used. They are related as
2
1 rpm = rad/s
60

4) Angular Acceleration:
The rate of change of angular velocity with respect to time is the angular
acceleration of the rotating body.
d
dt

  ……. [13.3]
The magnitude of angular acceleration is denoted by notation  (alpha)
and its direction acts along the axis of rotation. The sense of angular
acceleration is same as the sense of angular velocity, if the angular velocity
increases with time, and is opposite to the sense of angular velocity, if the
angular velocity decreases with time. Angular acceleration is represented
by curved arrow, representing clockwise or anticlockwise sense.
Units of angular acceleration are rad/s2

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Types of Rotation Motion about Fixed Axis
Rotation about fixed axis can be classified under three categories,
1) Uniform Angular Velocity Motion:
In this case of rotation motion the angular velocity of the rotating body
remains constant during motion. For such motions, we use a simple
relation relating , and t as 
.........[13.4]
t

 
2) Uniform Angular Acceleration Motion:
In this case of rotation motion, the angular velocity of the rotating body is
not constant, but increases or decreases at a constant rate.
0
0
If initial angular velocity
final angular velocity
angular acceleration
angular displacement of the body
t time interval, then
we relate , , ,




  





and t as
0= + a t  …… [13.5 (a)]
2
0= t + ½ a t  …… [13.5 (b)]
2 2
0= + 2 a   …… [13.5 (c)]
3) Variable Angular Acceleration Motion:
In this case of rotation motion the angular velocity changes during motion
and the rate of change of angular velocity is variable.
To solve problem on variable angular acceleration motion we make use of
the basic differential equations discussed earlier.
d
dt

  .........[13.2]
d
dt

  …… [13.2]
d
dt

  …… [13.3]
From equations 13.2 and 13.3, we have
d
d
 


 …… [13.6]

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Relation between Linear Velocity and Angular Velocity
 Consider a body rotating about a fixed axis passing through O.
 Let at the given instant, the angular velocity of the body be  rad/s
anticlockwise.
 All the particles on the rotating body will have the same angular velocity
but different linear velocity. If vP is the linear velocity of a particle P and
also rPO is the radial distance from P to O, then
P POv r  
 The sense of vP would be consistent with the direction of  .
 In general linear velocity v of any particle located at a radial distance r
from the axis of rotation O is related to the angular velocity  of the body
by a relation.
v r  ………. [13.7]
Relation between Linear Acceleration and Angular Acceleration
 Consider a body rotating about a fixed axis passing through O, having an
angular velocity  and angular acceleration  , at the given instant as
shown.
 We have studied in Chapter 9 that a particle
in curvilinear motion, has acceleration 'a'
which can be resolved into normal component
an and tangential component at. Similarly if
ap is the acceleration of particle P and if (an)p
and (at)p are its components, then

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 
 
2
2
po
n p
po
r ×ωv
a
r
 

   2
n pop
a r  
also  
 
 po
t po t pop p
d rdv dω
a r a r
dt dt dt



    
In general the linear acceleration a of any particle located at a radial
distance r from the axis of rotation O, has its components related to
angular velocity and angular acceleration by the relation
2
.na r  ………. [13.8 (a)]
.ta r a ………. [13.8 (b)]
Also total linear acceleration a = 2 2
n ta a

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Special case 1 of rotation motion: A block connected to s rotating
pulley
Consider a pulley of radius r, hinged at the centre and supporting a block by
a string wound over it. The rotational motion of the pulley is related to
translation motion of the block. This relation can be worked out.
Let at a given instant, the pulley have an angular position of  rad, angular
velocity of  rad/s and angular acceleration of 
rad/s2.
Let xA, vA and aA be the corresponding position, velocity
and acceleration respectively of the block A at this
instant.
Since the block translates, the string connecting it also
translates. We find point P on the pu1ley is common to
the rotating pulley and the translating block.
Refer figure. If point P belongs to the pulley, its
position, linear velocity and tangential acceleration is
given by
ps = r and pv = r and  t p
a r
Relating these parameters to the motion of the connected block, we have
A p Ax s x r   .……… [13.9 (a)]
A p Av v v r   ………. [13.9 (b)]
and  A t Ap
a a a r   ………. [13.9 (c)]
Equations 13.9 (a), (b) and (c) relate the motion of a block hanging from a
rotating pulley.

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 Special case 2 of rotation motion: A pulley coupled to another pulley
and they rotate without slip
Consider a pulley A of radius rA, be
coupled to another pulley B of radius rB,
and the two rotate without slip. The
rotational motion of pulley A is related to
the rotational motion of pulley B. This
relation can be worked out. Let at a given
instant, pulley A have an angular
position of A rad, angular velocity of A
rad/s and angular acceleration of A rad / s2.
Let B , B and B be the corresponding angular position, angular velocity
and angular acceleration of pulley B at this instant.
Since the pulley A rotates, it causes pulley B to also rotate. We find point P is
a common point to the two pulleys.
If point P belongs to pulley A, its position, linear velocity and tangential
acceleration is given by
P A AS = r θ and P A Av = r ω and t P A A(a ) = r α
Similarly point P also belongs to pulley B, therefore if the above parameters
are related to the pulley B, we have
P A A B Bs = r θ = r θ ………. (a)
P A A B Bv = r ω = r ω ………. (b)
t P A A B B(a ) = r α = r α ………. (c)
Equation (a), (b) and (c) relate the motions of two pulleys or gears engaged
with one another.

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EXERCISE 1
1. The tub of a washing machine is rotating at 60 rad/sec when the power is
switched off. The tub makes 49 revolutions before coming to rest.
Determine the constant angular deceleration of the tub and the time it
takes to come to a halt.
2. P2. A wheel rotating about a fixed axis at 15 rpm is uniformly accelerated
for 60 sec during which it makes 30 revolutions. Find
a) angular velocity in rpm at the end of the interval.
b) time required to attain a speed of 30 rpm. .
3. A point on the rim of a flywheel has a peripheral speed of 6 m/s at an
instant which is decreasing at a rate of 30 m/s2. If the magnitude of the
total acceleration of the point at this instant is 50 m/s2, find the diameter
of the flywheel.
4. A 1 m diameter flywheel has an initial clockwise angular velocity of 5 rad/s
and a constant angular acceleration of 1.5 rad/s2. Determine the number
of revolutions it must make and the time required to acquire a clockwise
angular velocity of 30 rad/s. Also find the magnitude of linear velocity and
linear acceleration of a point on the rim of the flywheel at t = 0.
5. A windmill fan during a certain interval of time has an angular acceleration
defined by a relation α = 18 e - 0-3 t rad/s2. The blades of the fan
describes a circle of radius 2.5 m. If at t = 0, ω = 0, determine at t = 5 sec
a) angular velocity of the fan
b) revolutions undergone by the fan.
c) Speed of the tip of fan blade.
6. A concrete mixer drum A is being rotated by two gears B and C. If the
concrete mixer is designed to attain a speed of 6 rad/s uniformly in 30 sec,
starting from rest and then maintain this speed, determine
d) the number of revolutions undergone
by the drum at t = 300 sec
e) the angular acceleration to be
developed in the driving gears.
f) velocity of concrete particles touching
the inner wall of the drum during uniform rotation motion of the
drum

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7. The variation of angular speed with time of a fan is shown. Find number of
revolutions undergone by the fan during a 10 minutes interval.
a) the angular acceleration and angular deceleration during this time
interval.
b) the magnitude of velocity and acceleration of a point on the tip of the
fan at t = 9 minutes, knowing that the fan described a circle of 1200
mm diameter.
8. The angular displacement of the rotating wheel is defined by the relation θ
= 1/4 t3 + 2 t2 + 18 rad. Determine the angular velocity and angular
acceleration of the wheel at t = 5 sec.
9. The angular acceleration of a rotating rod is given by the relation α = 9.81
cos θ -2.22 rad/s2. The rod starts from rest at θ = 0. Find
g) the angular velocity and angular acceleration of the rod at 0 = 30°.
h) the maximum angular velocity and the corresponding angle 0.
10. A belt is wrapped over two pulleys transmitting the motion without
slipping. If the angular velocity of the driver pulley A is increased uniformly
from 2 rad/s to 16 rad/s in 4 sec, determine
i) the acceleration of the straight position of the belt
j) the magnitude of total acceleration of a point on the rim of pulley B at
t = 4 sec.
k) the number of revolutions turned by the two pulleys at t = 4 sec.
11. Find the angular velocity in rad/s for
l) the second hand, the minute hand and hour hand of a watch,
m) the earth about its own axis.

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GENERAL PLANE MOTION
Translation motion and rotation about fixed axis motion are plane motions
since the motion of the body can be analysed by taking a representative slab
or a plane of the body. They may also be referred to as a Two Dimensional
Motion.
Any plane motion which does not fall under
the category of rotation about fixed axis or
translation motion can be put under the
category of general plane motion. In fact a
general plane motion is a combination of
translation motion and rotation motion. Three
examples of body performing general plane
motion are shown below.
In Fig. (a), Two blocks A and B travel is fixed slot performing translation
motion. The blocks are pin-connected by a link AB. The link AB moves from
position (1) to position (2) performing general plane motion. We observe that
the link AB rotates, but not about a fixed axis. Thus the centre of rotation of
the link AB performing general plane motion keeps on moving at every
instant.
In Fig. (b), rod AB, rod BC and block C, form a pin-connected mechanism.
The rod AB performs rotation motion about fixed axis at A. Piston C is free to
perform translation motion in the fixed slot. It is the rod BC which neither
performs pure translation or pure rotation motion. It is therefore said to
perform general plane motion. The centre of rotation of rod BC keeps on
changing as it performs general plane motion.

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In Fig. (c), a wheel rolls without slip on the
ground. The rolling wheel rotates as well as
translates. It therefore performs general plane
motion. In general any-body which rolls
without slip performs general plane motion.
Though G.P. bodies don’t actually translate
and then rotate in succession, but the motion can be duplicated by first
translating the body and then rotating it.
General Plane Motion = Translation Motion + Rotation Motion
Consider a body which has moved from position (1) to position (2) performing
G.P. motion. Refer Fig. Let P and Q be two arbitrary points chosen on it. We
may duplicate the motion by first translating the body to its position (2), i.e.
segment PQ maintains its orientation and remains parallel. Now we can
rotate the body about P to get the true orientation of the body. Hence G.P.
Motion is said to be a sum of Translation Motion and Rotation Motion.

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RELATIVE VELOCITY METHOD
 We know that a G.P. Motion is a sum of translation and rotation motion.
To find the angular velocity of a body performing G.P. Motion, we may use
the relative Velocity Method, which is one of the methods for analysing
G.P. motion. To understand this method we note down the required steps
and simultaneously take up an example and apply the procedure.
 Consider a rod AB pin-connected to pistons A and B which move in fixed
slots as shown. Let piston B have a known velocity vB to the right. Let us
learn to find angular velocity AB of the G.P. body AB.
Step 1. Locate a point on the G.P. body whose magnitude and direction of
velocity is known. Such a point is referred to as the reference point
and the known velocity is referred to as the translating velocity.
In our example, the velocity of point B i.e. vB is known. Hence B is
the reference point and vB is the translating velocity.
Step 2. Locate another point on the G.P body whose direction of velocity is
known. Such a point is referred to as translation point.
In our example, the direction of velocity of point A is known, since it
is constrained to move in the vertical slot. Point A is therefore the
translation point.
Step 3. Translate the body with the translating velocity and then rotate it
about the reference point with the angular velocity of the G.P body.
In our example, the bar is translated with translating velocity vB and
then rotated about reference point B with angular velocity AB .
Step 4. Write the relation for the relative linear velocity of translation point.
In our example, we write the relative linear velocity of translation
point A as
A/B AB ABV = r +  …… (1)

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Step 5. Write the relation for the absolute velocity of the
translation point. Simultaneously draw the vector
diagram for the same. Obtain the relative velocity of
the translation point and hence obtain the angular
velocity of the G.P body.
In our example,
A B A/BV = V + V ….. (2)
From relation (2), A
B/
V can be found out, which in turn when
substituted in relation (1) gives the angular velocity AB .
Instantaneous Centre Method
 Instantaneous Centre is defined as the
point about which the G.P. body rotates
at the given instant. This point keeps on
changing as the G.P. body performs its
motion. The locus of the instantaneous
centres during the motion is known as
centroid. Instantaneous Centre may be
denoted by letter I.
 Let us understand the Instantaneous Centre Method to find the angular
velocity of a G.P body. Let us work with the earlier example we took in
article 13.5.1
Given- vB i.e. velocity of block B
To find - Angular velocity of rod AB at given instant
Step 1. Locate a point on the G.P body whose magnitude, direction and
sense of velocity is known and another point whose direction of
velocity is known. Mark the direction of velocity (Dov) of these two
points.
In our example, the magnitude and Dov of point B (DovB) is known
and also Dov of point A is known (DovA)

16
Step 2. Draw perpendiculars to the direction of velocities (Dov) and extend
them to intersect at a point. Call this point as I.
Step 3. Point I i.e. the instantaneous centre, is the centre of rotation of the
G.P body at the given instant. Now treating the G.P body as a
rotating body about I, and using v = r ω relation, the angular
velocity of the G.P body can be found out.
In our example, the radial length rBI can be found out by geometry of
 ABI.
Next using vB = rBI × ωAB
The angular velocity ωAB can be found out
Also now knowing ωAB , velocity of block A can be found out, using
vA - rAI × ωAB

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ROTATION ABOUT FIXED POINT
In this type of rigid body motion, the body rotates about a fixed point, but the
axis of rotation passing through the fixed point is not stationary as its
direction keeps on changing. Such motion is a three dimensional motion.
Example of this type of motion is of a top rotating about the pivot at O. The
axis of rotation is not fixed but changes its direction as the top rotates about
fixed pivot point O. Refer Fig.
Another example of rotation about a fixed point is of the motion of a boom
which is ball and socket supported at a point O on the crane. Fig. 13.15
shows the boom of a crane being raised up by rotation about the z axis, and
at the same time the crane itself rotates about the y axis to position the boom
over its target.
GENERAL MOTION
 Any other motion of the rigid bodies which do not fit in any of the above
four types of rigid body motion, may be classified as a General Motion.
 Motion analysis of a rigid body having Rotation Motion about a Fixed Point
of General Motion of a rigid body are beyond the scope of this book.

18
EXERCISE 2
1. The rod ABC is guided by two blocks A and B which move in channels as
shown. At the given instant, velocity of block A is 5 m/s downwards.
Determine
a) the angular velocity of rod ABC
b) velocities of block B and end C of rod.
2. A rod AB 26 m long leans against a vertical wall. The end ‘A’ on the floor is
drawn away from the wall at a rate of 24 m/s. When the end ‘A’ of the rod
is 10 m from the wall, determine the velocity of the end ‘B’ sliding down
vertically and the angular velocity of the rod AB.
3. A slider crank mechanism is shown
in figure. The crank OA rotates
anticlockwise at 100 rad/s. Find
the angular velocity of rod AB and
the velocity of slider at B.
4. In a slider crank mechanism as
shown in figure the crank is
rotating at as constant speed of
120 rev/min. The connecting
rod is 600 mm long and the
crank is 100 mm long. For an angle of 30°, determine the absolute velocity
of the crosshead P.

19
5. Block ‘D’ shown in figure moves with a speed of 3 m/s. Determine the
angular velocities of links BD and AB and the velocity of point B at the
instant shown. Use method of instantaneous centre of zero velocity.
6. For the position shown, the angular velocity
of bar AB is 10 rad/s anticlockwise.
Determine the angular velocities of bars BC
and CD.
7. Rod BCD is pinned to rod AB at B and
has a slider at C which slides freely in
the vertical slot. At the instant shown,
the angular velocity of rod AB is 4 rad/s
clockwise. Determine
a) angular velocity of rod BD
b) velocity of slider C
c) velocity of end D of the rod BD
8. A rod AB 1.8 m long, slides against an inclined plane and a horizontal floor.
The end A has a velocity of 5 m/s to the right. Determine the angular
velocity of the rod and the magnitude of velocity of end B and the midpoint
M of the rod for the instant shown.

20
9. A slender beam AB of length 3 m which
remains always in a same vertical plane as its
ends A and B are constrained to remain in
contact with a horizontal floor and a vertical
wall as shown. Determine the velocity at point
B using instantaneous centre method.
10. A wheel of radius 0.75 m rolls without slipping on a horizontal surface to
the right. Determine the velocities of the points P and Q shown in figure
when the velocity of centre of the wheel is 10 m/s towards right.
11. One end of rod AB is pinned to the cylinder of
diameter 0.5 m while the other end slides
vertically up the wall with a uniform speed of 2
m/s. For the instant, when the end A is
vertically over the centre of the cylinder, find
the angular velocity of the cylinder, assuming it
to roll without slip.
12. A flanged wheel rolls to the left on a horizontal rail as shown. The velocity of
the wheel’s centre is 4 m/s. Find velocities of points D, E, F and H on the
wheel.

21
13. Rod AB has a constant angular velocity of 50
rpm clockwise. For the given position of the
mechanism, find the angular velocity of rods BC
and CD.
14. If the link CD is rotating at 5 rad/sec anticlockwise, determine the angular
velocity of link AB at the instant shown.
15. Blocks A, B and C slide in fixed slots as shown.
The blocks form a mechanism, being
interconnected by pin-connected links AB and
BC. L (AB) = 400 mm and L (BC) = 600 mm. At
the given instant, block A has a velocity of 0.15
m/s downwards. Determine the velocities of
blocks B and C for the given instant.
16. In the mechanism shown the angular
velocity of link AB is 5 rad/s anticlockwise.
At the instant shown, determine the
angular velocity of link BC, velocity of
piston C and velocity of midpoint M of link
BC.
17. In the engine system shown, the crank AB has a constant clockwise
angular velocity of 2000
rpm. For the crank
position shown determine
the angular velocity of
connecting rod BD and
velocity of the piston D. Use ICR method.

22
18. Locate the instantaneous center of rotation
for the link ABC and determine velocity of
points B & C. Angular velocity of rod OA is
15 rad/sec counter clock wise. Length of OA
is 200 mm, AB is 400 mm and BC is 150
mm.
19. The bar BC of the linkage shown has an angular velocity of 3 rad /sec
clockwise at the instant shown. Determine the angular velocity of bar AB
and linear velocity of point P on the bar BC.

23
EXERCISE 3
Theory Questions
Q.1. Classify types of motion for rigid body with suitable examples.
Q.2. Explain the characteristics of a Translating Body.
Q.3. For a rotating body explain in brief the terms
1) Angular Position,
2) Angular Displacement
3) Angular Velocity
4) Angular Acceleration.
Q.4. What sire the different types of Rotation Motion about fixed axis and list
the corresponding equations applicable?
Q.5. Explain Instantaneous Centre of Rotation.
Q.6. Write a short note on General Plane Motion.
UNIVERSITY QUESTIONS
1. For the link and slider mechanism shown in figure, locate the
instantaneous centre of rotation of link AB. Also find the angular velocity of
link ‘OA’. Take velocity of slider at B = 2500mm/sec. (10 Marks)
2. ‘C’ is a uniform cylinder to which a rod ‘AB’ is
pinned at ‘A’ and the other end of the rod ‘B’ is
moving along a vertical wall as shown in fig. If the
end ‘B’ of the rod is moving upward along the wall
at a speed of 3.3 m/s find the angular velocity of
the cylinder assuming that it is rolling without
slipping. (6 Marks)

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3. Find the forces in members BD, BE and CE by
method of section only for the truss shown in
the fig. Also find the forces in other members
by method of joints. (8 Marks)
4. Collar B moves up with constant velocity
BV =2m/s. Rod AB is pinned at B.
Find out angular velocity of AB and velocity of A. (6 Marks)
5. Fig. shows the crank and connecting rod mechanism. The crank AB rotates
with an angular velocity of 2 rad/sec in clockwise direction. Determine the
angular velocity of Connecting Rod BC and the velocity of piston C using
ICR method. AB = 0.3 m and CD = 0.8 m (6 Marks)
6. In the mechanism shown the angular velocity of link AB is 5 rad/sec
anticlockwise. At the instant shown, determine the angular velocity of link
BC and velocity of piston C. (6 Marks)

25
7. Due to slipping, points A and B on the rim of the disk have the velocities as
shown in figure. Determine the velocities of the centre point C and point D
on the rim at this instant. Take radius of disc 0.24 m (6 Marks)

Kinematics of rigid bodies

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Kinematics of rigid bodies