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
`[ML`

, and its SI unit is newton-metre ^{2}T^{-2}]`(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:

```
τ
```

_{ext} = Iα

In the above expression, `τ`

is the torque associated with external forces acting on the body, I is the _{ext}**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

.
Let the angle between the vectors **p** = m**v****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.

Kinematics Of Rotation About A Fixed Axis
59:52
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Torque Of A Force Acting On A Particle
44:39
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Torque Of Coplanar Forces And Of A Couple
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Equation Of Rotational Motion Of A Rigid Body
44:58
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More Problems On Equation Of Rotational Motion Of Rigid Body
47:57
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Moment Of Inertia, And Its Calculation For Various Bodies I
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Calculation Of Moments Of Inertia For Various Bodies II
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Two Important theorems On Moment Of Inertia
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Applications Of Parallel And Perpendicular Axes Theorems
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Problems On Moments Of Inertia Of Various Bodies
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Problems On Moments Of Inertia Of Various Bodies II
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Overturning Of A Vehicle At A Bend
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Angular Momentum Of A Particle And Its Relation With Torque
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Problems On Angular Momentum Of A Particle
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More Problems On Angular Momentum
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Angular Momentum Of System Of Particles And Angular Impulse
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Problems on angular impulse
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Principle Of Conservation Of Angular Momentum, And Its Applications
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Problems On Principle Of Angular Momentum Conservation
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Work And Power In Rotational Motion
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Rotational Kinetic Energy, And Comparison Between Translation And Rotation
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Rolling Motion Of A Rigid Body On Horizontal Floor
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Problems On Rolling Motion On Horizontal Floor
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Rolling Motion Of A Rigid Body Down An Incline
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Problems On Rolling Motion Down An Incline
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A Cylinder Falling At The End Of An Unwinding String
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Problems On Rolling Motion At The End Of Unwinding String
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Problems On Rotating Pulley Of Non-Negligible Mass
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Combined Translational And Rotational Motion As Pure Rotation
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Advanced-Level Problems On Rotational Mechanics I 1:03:47

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Advanced-Level Problems On Rotational Mechanics II 55:28

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Advanced Level Problems On Rotational Mechanics III 01:41:07

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Advanced Level Problems On Rotational Mechanics IV 01:18:23

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Advanced Level Problems On Rotational Mechanics V 01:19:03

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Advanced Level Problems On Rotational Mechanics-VI 01:28:18

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Advanced Level Problems On Rotational Mechanics VII 01:21:12

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Advanced Level Problems On Rotational Mechanics VIII 01:38:01

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Advanced Level Problems On Rotational Mechanics IX 01:14:15

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Advanced Level Problems On Rotational Mechanics X 45:34

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Problems On Rotational Mechanics I (CE) 1:04:17

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Problems On Rotational Mechanics II (CE) 1:12:42

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Problems On Rotational Mechanics III (CE) 1:06:48

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Problems On Rotational Mechanics IV (CE) 1:04:12

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