Rotational motion

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Topic updated on 10/13/2020 03:25pm

A body is said to perform a pure rotational motion if every particle in the body moves in a circular path such that the centers of all those circles lie on a single straight line called as the axis of rotation.

Angular acceleration

When an object rotates its angular velocity changes with time.Angular acceleration is defined as rate of change of angular velocity.

Image result for angular acceleration formula

[α]=T^-2 Unit – rads^-2

The translational acceleration of a point on the object rotating is given by

a_T=rα

where r is the radius or distance from the axis of rotation. This is also the tangential component of acceleration: it is tangential to the direction of motion of the point.

When the angular acceleration is constant, the five quantities angular displacement $\theta$ , initial angular velocity $\omega _{i}$ , final angular velocity $\omega _{f}$ , angular acceleration $\alpha$ , and time $t$ can be related by four equations of kinematics:

\omega _{f}=\omega _{i}+\alpha t\;\!

\theta =\omega _{i}t+{\begin{matrix}{\frac {1}{2}}\end{matrix}}\alpha t^{2}

\omega _{f}^{2}=\omega _{i}^{2}+2\alpha \theta

\theta ={\tfrac {1}{2}}\left(\omega _{f}+\omega _{i}\right)t

Moment of inertia

The moment of inertia of an object is a measure of the object’s resistance to changes to its rotation. The moment of inertia is measured in kg m². It depends on the object’s mass: increasing the mass of an object increases the moment of inertia. It also depends on the distribution of the mass: distributing the mass further from the centre of rotation increases the moment of inertia by a greater degree.

For a single particle of mass m a distance r from the axis of rotation, the moment of inertia is given by I=mr²

Torque

Torque ${\boldsymbol {\tau }}$ is the twisting effect of a force F applied to a rotating object which is at position r from its axis of rotation.

{\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} ,

A net torque acting upon an object will produce an angular acceleration of the object according to

{\boldsymbol {\tau }}=I{\boldsymbol {\alpha }},

just as F = ma in linear dynamics.

The work done by a torque acting on an object equals the magnitude of the torque times the angle through which the torque is applied

W=\tau \theta .\!

The power of a torque is equal to the work done by the torque per unit time

P=\tau \omega .\!

Angular momentum

The angular momentum L is a measure of the difficulty of bringing a rotating object to rest. It is given by

L = mv×r

Angular momentum is related to angular velocity by

{\mathbf {L}}=I{\boldsymbol {\omega }},

just as p = m v in linear dynamics.

The greater the angular momentum of the spinning object such as a top, the greater its tendency to continue to spin.

The Angular Momentum of a rotating body is proportional to its mass and to how rapidly it is turning. In addition the angular momentum depends on how the mass is distributed relative to the axis of rotation: the further away the mass is located from the axis of rotation, the greater the angular momentum . A flat disk such as a record turntable has less angular momentum than a hollow cylinder of the same mass and velocity of rotation.

Like linear momentum, angular momentum is vector quantity, and its conservation implies that the direction of the spin axis tends to remain unchanged. For this reason the spinning top remains upright whereas a stationary one falls over immediately.

Torque and angular momentum are related according to

{\boldsymbol {\tau }}={\frac {d{\mathbf {L}}}{dt}},

just as F = dp/dt in linear dynamics

Law of conservation of angular momentum

If there is no external torque acting on a system the angular momentum remains unchanged.

L=Iω=constant

Rotational kinetic energy

When an object rotates about an axis every particle in that body moves in a circle then rotational kinetic energy is the sum of total kinetic energy of particle.

Rotational kinetic energy = ½Iω²

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