Relative Motion

Suppose two cars are moving at the same velocity. Even though they are moving with respect to an observer on the road, they are not moving with respect to each other. A passenger in one car will see the other car at the same distance. Thus, with respect to time, the other car is not having any velocity. We say that relative motion of one car with respect to other car is zero. We take up this problem in a general way.

Supposing an axes system is moving with respect to other axes system, what is the relationship between velocity in two system?

X-Y is a fixed reference frame and x-y is a moving reference frame. To begin with, consider the x-y axes only translate with respect to X-Y, but do not rotate. If A is any particle. The position Vector of A as measured relative to the frame x-y is , where subscript A/B means "A relative to B" or "A with respect to B". The position of A with respect to X-Y,

131 Differentiating it,

or,

Thus, the absolute velocity of a particle is the vector sum of the velocity of a particle with respect to a translating frame of reference and the velocity of the frame.

Similarly,

or,

The absolute acceleration of a particle is the sum of the acceleration of the particle with respect to the translating frame and the acceleration of the frame.

Also,

Note that have zero derivatives with respect to time as these are constant vectors. Their direction and magnitude both remain constant with respect to time. They only translate.

A translating reference frame which has no acceleration is known as inertial frame.

An

example: A card board is moving in plane floor with a velocity with respect to some fixed X-Y axes. A particle is rotating in a circle of radius 2 units with a speed of 5 rad/sec. What is the absolute (with respect to X-Y) velocity of particle when the particle is at P?

Solution:

132 Let us fix a x-y axis system at the center of the circle.

With respect to x-y system velocity of P = =

Thus the absolute velocity of P =

With respect to x-y system, acceleration of the particle = -50 units(towards centre) The acceleration of x-y with respect to X-Y system is zero.

Hence the absolute acceleration is .

Let us consider axes system xy which rotates with respect to XY.

Angular Velocity  =

Now

133 and

parent from the following figure.

If angular velocity is denoted by the vector

Multiple Choice Question

Q.1. The velocity of a particle is in a frame of reference rotating with an angular velocity of10 rad/sec, i.e., unit vector and rotate with an angular velocity of 10 rad/sec about an axis perpendicular to and . The acceleration of the particle at t =25 is

a. 5 + 24 b. -235 + 124 c. 235 + 124 d. 235 - 124

Q.2. A unit vector is rotating about an axis making an angle of 45⁰ with the unit vector. The angular speed is 10 rad/sec. The magnitude of the rate of change of the unit vector is

a. 10 rad/sec b. 5 rad/sec c. 7.07 rad/sec d. 1.41 rad/sec

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Q.3. A table is rotating at an angular speed of 1 rad/sec. An ant starts moving on the table radially outwards at a speed of 2 mm/sec. At the instant when the ant is 5 mm from the center, the magnitude of its velocity as observed by a fixed outside observer is

a. 2 mm/sec b. 5 mm/sec c. mm/sec d. mm/sec

Q.4. A boy is enjoying a merry-go-round, which is rotating at an angular speed of 2 rad/sec. The boy is at a distance of 1 m from the axis of the revolution.

When the body is at position A, another boy approached towards A, running at a speed of 0.01 m/sec. The magnitude of the velocity of walking boy as seen by the revolving boy is

a. 0.01 m/sec b. 2 m/sec c. 2.1 m/sec d. 1.1 m/sec

Q.5. A slider moves with a velocity 20 m/sec along a link rotating at 10 rad/sec. The magnitude of the coriolis component of the acceleration is

a. 200 m/sec² b. 100 m/sec² c. 400 m/sec² d. 20 m/sec²

135 Q.6.

A hub with an attached blade rotates about a vertical axis. The front viewer (projection in vertical plane) is shown. If the blade vibrates in vertical plane, there will be

a. a coriolis component of acceleration

b. a coriolis component of acceleration in the vertical plane

c. a coriolis component of acceleration in the vertical plane and a coriolis component of acceleration in the horizontal plane

d. no coriolis component of Q.7.

A hue with an attached blade rotates in a vertical plane. The top view (projection) in the horizontal plane is shown. If the blade vibrates, there will be

a. no coriolis component of acceleration

b. a coriolis component of acceleration along vertical axis c. a coriolis component of acceleration along radial direction

d. a coriolis component of acceleration along the direction of vibration.

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Q.8. A particle moves with constant relative velocity vr on the periphery of a disc of radius r in the clockwise direction. The disk also rotates with an angular velocity in the clockwise direction.

The absolute acceleration of the particle is

a.

b.

c. d. none of these.

Q.9. A link is rotating with an angular speed of . A particle slides on the link with a velocity of v.

If the direction of rotation of link is changed, but the speed remains same, then a. the magnitude and direction of the acceleration remains same.

b. the magnitude of the acceleration remains same, but the direction changes.

c. the magnitude of the acceleration changes, but the direction remains same.

d. both the direction and magnitude of the acceleration changes.

Q.10.

The velocity of particle A relative to B is

a. 3 c.

b. d. none of these.

137 PROBLEMS

.1. A disk of radius r is having the angular velocity and an angular acceleration of . A

particle P moves in the opposite direction around the circumference with uniform relative velocity v_r. Find the absolute acceleration of P.

Q.2.

In the figure, if the crank rotates with an angular speed , find out the acceleration of point P with respect to C, where C is the mid-point of crank AB.

Q.3.

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A disk is rotating at an angular speed . A particle starts in the y direction at a speed v. Find out the absolute acceleration of the particle when it reaches the periphery.

Q.4. A car is approaching at a velocity v. Find the acceleration of car relative to A.

Q A link is rotating with an angular velocity of 10 rad/sec and angular acceleration of 1 rad/sec². A particle slides on the link with velocity of 10 m/sec and acceleration of 1 m/sec². At the time when the particle is at a distance of 10 cm from the center of rotation of link, find out the absolute acceleration of the particle. Also, mention what are the various components of the acceleration.

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