Rotational Motion of a Rigid Body

Q. 1. What is meant by a rigid body ?

Ans.Rigid Body : "A rigid body is a body which consists of enormous number of particles closely and rigidly connected together in such a way that the distance between any two constituent particles of it remains unchanged under every external force." If a rigid body undergoes some displacement, then every constituent particle of the body undergoes the same displacement. If the body rotates through a certain angle about an axis every particle in it rotates through the same angle about that axisand every point in the body moves in a circle whose centre is on the axis and which lies in a plane perpendicular to the axis.

Q. 2. How many kinds of motions can a rigid body have ? Define general motion of a rigid body. 1

Ans.Motion of a Rigid Body: Generally a rigid body can possess two kinds ofmotion namely (i) translatory motion and (ii) rotatory motion (i) Translatory Motion: "A rigid body is said to have a pure translational motion if every particle of the body undergoes same displacement in the same direction during any time interval."

I like to share this Physics Equations of Motion with you all through my article.

When a body is displaced from one place to other without rolling, then motion of the body is translatory motion.

Supposea bicycle is being pushed along the rod with brakes applied, then everyparticle of the bicycle undergoes same linear displacement parallel to the rod, therefore such motion of the bicycle is pure translatory motion.

(ii) Rotational Motion : "A body is said to have a pure rotational motion if every particle of the body move on circular paths and the centres of all such circles traced by these particles lie on a fixed straight line called axis of rotation."

For example when the wheel of a bicycle is rotated, keeping the bicycle stationary, every particle of the wheel move on circular path with centre of circle lying on the axis of'rotation of wheel.. So in such case, motion of the wheel is a purely rotational.

(iii)General Motion of a Rigid Body : Generally every motion of a free rigid body is composed of both kinds of motion, translational motion as well as rotational motion. Such a motion of rigid body is called generalmotion of the rigid body.

Whena wheel is rolled on a surface, every particle of the wheel moves on a circular path with centre at the axis of wheel while the wheel itself moves in forward ' direction. Thus wheel performs rotational as well-as translational motion, so this motion is the general motion of the wheel.

Q. 3. What do you understand by angular acceleration ? Write its unit and dimensional formula.

Ans.Angular Acceleration : The rate of change of angular velocity of a rigid body about the axis of rotation is called the angular accelerationof that body. It is denoted by a. Let the angular velocity of body rotating about an axis changes from o to co + Aco in time interval At, then average angular acceleration during this interval is given by,

a = [a denotes the average angular acceleration ]

If time interval Af is very small (Af -> 0) then average angular acceleration becomes instantaneous angular acceleration.

.'. Instantaneous angular acceleration is T . Aco da a = Lim — = — At ->o Af dt

The unit of angular acceleration is radian/second2 and it's dimensional formula is [T ~2].

Other Expressions For Angular Acceleration: If body rotates through an angle 0

in time interval t, then instantaneous angular velocity co is given by,

dQ a =— dt . . . ^ da d2& :. Angular acceleration a = — = , d(£> dm dQ da and a = — = ----- — co _dt dQ dt dQ_

a ' i i ^ da d2Q dQ Angular acceleration a == — = —= co — 6___dt dt2 dt

Q. 4. Establish a relation between linear acceleration and angular acceleration.

Ans.Relation Between Linear and Angular Acceleration : Let a body be rotating about an axis. Let a particle inside the body'be at a distance rfrom the axis of rotation. Let at time t, this particle has angular velocity co and linear velocity v then we have, v = m ...(1)

If at time t + At this particle has angular velocity co + Aco and linear velocity v + Av v + Av = r (co + Aco) ...(2)

(v Body is rigid r="constant)" Av = (v + Av) - v = r (co + Aco) - r co

or change in linear velocity Av = r Aco

T . Av T . Aco Lim — = Lim r — Af->0 Af Af->0 At ,

But Lim — is instantaneous linear accelerations and Lim — is instantaneous A<->0 Af A*-»0 Af '

angular acceleration a. .". We have | a = r a |

or linear acceleration = distance of the particle from the axis

x angular acceleration This is the required relation.

Inthe case of a rigid body, during any time interval every particle inside the body undergoes same angular displacement and therefore every particle has same angular velocity at given instant and similarly every particle has same angular acceleration at given instant, but their linear accelerations are different (as shown by relation obtained) whichare proportional to their distances from the axis of rotation.

Q. 5. What do you understand by moment of force (torque) ? Write its unit and dimensional formula. (U.A.2005)

Ans.Moment of Force (Torque) : The tendency of an external force to rotate abody about an axis is called the moment of thatforce or torque about that axis.

Thetendency of force to rotate a body not only depends upon the force but also on the distance of the line of action of the force from the axis ofrotation, so the moment of a force about ah axis is defined as the product of the magnitude of the force (F) and the perpendicular distancep of the line of action of force from the axis of rotation. Moment of force is denoted byx. /. By definition z = Fp ...(1

where p is length of perpendicular drawn on the line of action of force from the axis of rotation.

If r is the position vector of a point on the line of action of force with respect to a -»

pointon the axis of rotation and F represents the force in vector form. Thenin vector notation the moment of force is given by r = r x F| ...(2)

Physics is widely used in day to day activities watch out for my forthcoming posts on Enthalpy of Sublimation and Centripetal Force Equation. I am sure they will be helpful.

Ifforce has a tendency to rotate the body in anticlockwise direction, then moment of force is takenpositive, but if it has a tendency to rotate in clockwise direction then moment of force is taken negative. Unit of Torque : Since in M.K.S. system the unit of length is metre and unit of force is newton, hence in M.K.S. system the unit of torque is newton-metre. Dimensional Formula:Torque x = Fp :. Dimensional formula of torque =[MLT~2][L]

= [ML2T"2]

From eq. (1) we conclude that:

(i)If the line of action of the force is passing through the axis of rotation (p = 0), then moment of force will be zero. In that case force will be incapable to rotate the body, ho we ver intense the force may be.

(ii) If the line of action of the force is moved away from the axis of rotation, then momentof force will be increased. Then it will become easier to rotate the body. That is why, the handle of the screw is made wider and handle or nob is provided on the free edge of the door (farthest from the axis of rotation).