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In addition to the revision notes for Dynamics of Rotational Motion on this page, you can also access the following Rotation learning resources for Dynamics of Rotational Motion

Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
---|---|---|---|---|---|---|

7.2 | Dynamics of Rotational Motion |

In these revision notes for Dynamics of Rotational Motion, we cover the following key points:

- The relationship between force and torque in translational and rotational motion respectively
- The meaning of moment of inertia and how to calculate it in various cases
- How to write the Newton's Second Law in rotational motion
- The concept of angular momentum and its relevant equation
- Work, kinetic energy and power in rotational motion and how they are related between each other and their analogue translational quantities

**Moment of force** in rest and especially **torque** when the system is in motion is the rotational equivalent of **force**. The equation of torque is

τ*⃗* = r*⃗* × F*⃗*

**Moment of inertia**, I in the rotational motion is the analogue of **mass**. It is a quantity expressing a body's tendency to resist angular acceleration, i.e. to change in the actual state of rotation. The general equation of moment of inertia is

I = m × r^{2}

The unit of moment of inertia is [kg × m2].

The above equation can take other forms based on the structure and shape of the rotating object involved. Thus,

**a.** Moment of inertia of a bar of mass m and length L when rotating around its centre is:

I = *1**/**12* m × L^{2}

**b.** Moment of inertia of a bar of mass m and length L when rotating around its end is:

I = *1**/**3* m × L^{2}

**c.** Moment of inertia of a cylinder or disc of mass m and base radius R when rotating around its axis of symmetry, which passes along the height, is:

I = *1**/**2* m × R^{2}

**d.** Moment of inertia of a cylinder or disc of mass m, height h and base radius R when rotating around its central diameter is:

I = *1**/**4* m × R^{2} + *1**/**12* m × h^{2}

**e.** Moment of inertia of a ring of mass m and radius R when rotating around its axis of symmetry, which passes along the centre, is:

I = m × R^{2}

**f.** Moment of inertia of a ring of mass m and radius R when rotating around its diameter is:

I = *1**/**2* m × R^{2}

**g.** Moment of inertia of a sphere of mass m and radius R when rotating around its central diameter is:

I = *2**/**5* m × R^{2}

**h.** Moment of inertia of a spherical shell of mass m and radius R when rotating around its central diameter is:

I = *2**/**3* m × R^{2}

Newton's Second Law of Motion in Linear Dynamics is F = m × a. Thus, giving the analogy between translational and rotational quantities, we can write the Newton's Second Law of Motion in rotational motion as

τ = I × α

where τ is the torque, which is analogue to force in linear dynamics and α is the angular acceleration.

**Angular momentum**, L, is a vector quantity (more precisely, a pseudo-vector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. It is the equivalent of **momentum** in linear motion.

The equation of angular momentum is

L = I × ω

and its unit is [kg × m2 / s].

In the rotational world, work, W is done by a torque τ applied through some angle φ. Mathematically, we have

W_{rot} = τ × φ

Since moment of inertia is analogue to mass and angular velocity is analogue to linear velocity, we obtain for kinetic energy of rotational motion:

KE_{rot} = *I × ω*^{2}*/**2*

Power in rotational motion is

P_{rot} = τ × ω

because power in translational motion is P = F × v.

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