LEC. (1)-Kinematics of Rigid Bodies-Definitions-Translation-Rotational Motion-Examples

8/2/2019 LEC. (1)-Kinematics of Rigid Bodies-Definitions-Translation-Rotational Motion-Examples

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Kinematics of Rigid Bodies

Plane Kinematics of Rigid Bodies

What is meant by a Rigid body:

It is a system of infinite number of particles having constant distance between anytwo arbitrary partices irrespective the forces acting on the body. So, the distance

AB remains constant if the body moves from the position at the instant t1 to the

position at the instant t2. As a result all the three angles involved by any triangle

ABC in the rigid body also remain the same during the motion and all lines ( such

AB , BC , CA ) rotate the same angle and in the same direction. The velocity at

which these lines rotate us called ( The angular velocity of the rigid body).

Rigid body motion forms :

1.TRANSLATION (Parallel motion):In such a motion , all the lines in the body ( such AB) moves with a constant

direction ( does not rotate) so A1 B1 B2 A2 is a parallelogram and the

displacements ot all the points ( A, B) are equals in magnitudes and have

the same directions. The points of the rigid body either move in straight

lines and the translation in this case is called rectilinear translation , or

A

BC

t1

A

B

C

t2


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2ca

a

B

A

2b

B

A

2a

Br

Ar

A

B

A

B

move along curvilinear paths and the translation in this case is called

curvilinear translation.

Velocity and Acceleration in Translation :

BA rr , tr

t

r BA

, =BA , aaa BA

1a: rectilinear

translation

1A

1B

2A

2B

1A

1B

2A

2B

1b:curve-linear

translation


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3a

O

1A

2A

1B 2B

O

1A

2A

1B 2B

3b

2.Rotation about affixed axis:

All the points ( A ,B , ) move along concentric circles of common centre O at

the fixed centre of rotation. The fixed centre of rotation may be in the rigid body

itself ( Fig. 3a) or outside the rigid body ( Fig. 3b) . In all cases the centre of

rotation lies in the plane of motion.

Angular Velocity:

dt

d Angular acceleration :

d

d

dt

d

dt

d2

2

O Q

P

3c

O

O

3d

Accelerating rotation

Decelerating rotation

The direction

of rotation is

anticlockwise


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The Velocity of a point :

rs , rdt

dr

dt

ds , r

The acceleration of a point :The normal Component :

Oinrar

ra

2n

222

n

The tangential Component :

onacceleratiangulartheO withaboutrotatesandatolarperpendicu(

rradt

dr

dt

sda

n

t2

2

2

2

t

)(tan,C

,ka

atan,rCr)(aaa

2

124

2n

t242n

2t

a

P

4 a

Qr s

O

Pr

rat

O

ra2

n P

4c4 b

r

O

P

O


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Rolling without slipping between two moving surfaces:

The sufficient and necessary conditions for the rolling without slipping between

two moving surfaces:

The tangential components of the velocities and the

tangential components of the accelerations the points of

tangency on both surfaces are equal in magnitude and having the same direction:

Applications :

Pulley ropeweight system:

Velocity relations:

a1 v1

v2

a2

v1T , a1T

v2T , a2T

Common tangent

A

B C


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Acceleration relations:

Example:

A

B C

0.5 m 1.5 m

A

B

O

For the given pulley rope weight system ,

the weight A has the shown velocity and

acceleration. Determine the angular velocity

and angular acceleration of the pulley (O),

and the velocity and acceleration of the

weight (B).


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velocity analysis

0.5 m 1.5 m

A

B

O

0.5 m 1.5 m

A

B

O

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LEC. (1)-Kinematics of Rigid Bodies-Definitions-Translation-Rotational Motion-Examples