# Rotational Mechanics

There are many instances of rotation around us. As a rotating body moves from one position to another, a straight line joining any two points on the body makes an angle with its previous position. This is an important criterion which helps to distinguish between rotation and certain kinds of curvilinear motion. In the first part, we discuss rotation about a fixed axis, otherwise known as pure rotation. The motion of a fan blade, a gramophone record, the steering wheel of a car etc is of this type. The second part deals with rotation about a moving axis which, in effect, is a combination of translation and rotation. A sphere rolling on the floor or a yo-yo bouncing up and down performs this kind of motion.

When a rigid body performs pure rotation, one line in the body (or body extended) remains fixed in space while all points not lying on the line move in circles, each having its centre on the fixed line. We call this line the axis of rotation. Because of this close connection between circular motion and rotational motion, the kinematic equations of circular motion developed earlier are also valid for rotation of a rigid body. As you will recall, the equations involve quantities such as angular displacement, angular velocity and angular acceleration.

The rotational effect of a force depends not only on its magnitude, but also on the position of its line of action. The effect is measured in terms of a physical quantity called torque or moment of a force. Suppose P is the position of a particle in the xy-plane. The line of action of the force F acting on the particle lies in the same plane. The position vector of P with respect to the origin O is r. Let the angle between the vectors r and F be θ. The torque τ of F about O is defined as the cross product of r and F. That is:

`τ = r × F`

The magnitude of the torque vector is given by:

` τ = rF sinθ = Fd`

where `d = r sinθ` is the perpendicular distance from O to the line of action of F. The quantity d is known as the moment arm of F about O. The direction of the torque follows from right-hand corkscrew rule. The dimensions of torque are `[ML2T-2]` , and its SI unit is newton-metre `(N-m)`.

Two equal and opposite forces having parallel lines of action form a couple. While the resultant of these two forces is zero, the net torque produced by them about any point is never zero.

The equation of rotational motion of a rigid body is:

``` τext = Iα ```

In the above expression, `τext` is the torque associated with external forces acting on the body, I is the moment of inertia of the body about the axis of rotation and α is its angular acceleration. Recognize that this equation is the rotational analogue of `F = ma`, applied to the rectilinear motion of a particle.

A detailed study of moment of inertia will take us through the definition of radius of gyration and calculation of moment of inertia of various bodies by the method of integration. Two useful theorems on moment of inertia – parallel axes theorem and perpendicular axes theorem – will also be discussed.

Just like linear momentum is important in the discussion of translational motion, angular momentum is important in the discussion of rotational motion. Consider a particle of mass m moving in the xy-plane. At some instant, the particle is at the point P whose position vector with respect to the origin O is r. The velocity of the particle at this instant is v, and therefore, its linear momentum is `p = mv`. Let the angle between the vectors r and p be θ. The angular momentum, l, of the particle about O is defined as the cross product of r and p. That is:

`l = r × p`

The magnitude of the angular momentum vector is given by:

``` l = rp sinθ = mvr sinθ ```

The direction of l is given, as usual, by the right-hand corkscrew rule.

According to the principle of conservation of angular momentum, the total angular momentum of a system of particles remains constant if the net external torque acting on the system is zero.

So far we discussed rotation of a rigid body about a fixed axis. We may now consider more general cases where the axis of rotation moves in space. Take the case of a wheel rolling on a horizontal straight road. An observer standing on the ground sees that the wheel rotates about a horizontal axis through the centre of mass and, at the same time, the centre of mass itself moves along a straight line. So this becomes a case of combined translational and rotational motion. A few cases worth analysing are rolling motion of a rigid body on horizontal floor and down an incline, a cylinder falling at the end of an unwinding string etc.

#### Basic level videos

Kinematics Of Rotation About A Fixed Axis 59:52 Basic
300 5
Torque Of A Force Acting On A Particle 44:39 Basic
250 4
Torque Of Coplanar Forces And Of A Couple 44:48 Basic
250 4
Equation Of Rotational Motion Of A Rigid Body 44:58 Basic
250 4
More Problems On Equation Of Rotational Motion Of Rigid Body 47:57 Basic
300 5
Moment Of Inertia, And Its Calculation For Various Bodies I 56:08 Basic
300 5
Calculation Of Moments Of Inertia For Various Bodies II 49:23 Basic
300 5
Two Important theorems On Moment Of Inertia 30:42 Basic
200 3
Applications Of Parallel And Perpendicular Axes Theorems 46:50 Basic
300 5
Problems On Moments Of Inertia Of Various Bodies 41:14 Basic
250 4
Problems On Moments Of Inertia Of Various Bodies II 53:40 Basic
300 5
Overturning Of A Vehicle At A Bend 53:58 Basic
300 5
Angular Momentum Of A Particle And Its Relation With Torque 57:10 Basic
300 5
Problems On Angular Momentum Of A Particle 1:03:24 Basic
350 7.5
More Problems On Angular Momentum 59:10 Basic
300 5
Angular Momentum Of System Of Particles And Angular Impulse 32:23 Basic
300 3
Problems on angular impulse 43:31 Basic
250 4
Principle Of Conservation Of Angular Momentum, And Its Applications 34:47 Basic
200 3
Problems On Principle Of Angular Momentum Conservation 1:01:03 Basic
350 7.5
Work And Power In Rotational Motion 28:46 Basic
200 3
Rotational Kinetic Energy, And Comparison Between Translation And Rotation 58:40 Basic
300 4
Rolling Motion Of A Rigid Body On Horizontal Floor 1:08:46 Basic
350 7.5
Problems On Rolling Motion On Horizontal Floor 1:05:44 Basic
350 7.5
Rolling Motion Of A Rigid Body Down An Incline 49:48 Basic
300 5
Problems On Rolling Motion Down An Incline 1:12:27 Basic
350 7.5
A Cylinder Falling At The End Of An Unwinding String 41:25 Basic
250 4
Problems On Rolling Motion At The End Of Unwinding String 49:40 Basic
250 4
Problems On Rotating Pulley Of Non-Negligible Mass 37:59 Basic
200 3
Combined Translational And Rotational Motion As Pure Rotation 58:52 Basic
300 5

#### Advanced level Videos Note: (CE) Stands for Problems from Competitive Examination Papers

Advanced-Level Problems On Rotational Mechanics I 1:03:47
350
7.5
Advanced-Level Problems On Rotational Mechanics II 55:28
300
5
Advanced Level Problems On Rotational Mechanics III 01:41:07
450
8
Advanced Level Problems On Rotational Mechanics IV 01:18:23
350
7.5
Advanced Level Problems On Rotational Mechanics V 01:19:03
350
7.5
Advanced Level Problems On Rotational Mechanics-VI 01:28:18
350
7.5
Advanced Level Problems On Rotational Mechanics VII 01:21:12
350
7.5
Advanced Level Problems On Rotational Mechanics VIII 01:38:01
450
8
Advanced Level Problems On Rotational Mechanics IX 01:14:15
350
7.5
Advanced Level Problems On Rotational Mechanics X 45:34
250
4
Problems On Rotational Mechanics I (CE) 1:04:17
350
7.5