Dynamics of Rotational Motion

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7.2Dynamics 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

Dynamics of Rotational Motion Revision Notes

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 × r2

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 × L2

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

I = 1/3 m × L2

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 × R2

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 × R2 + 1/12 m × h2

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 × R2

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

I = 1/2 m × R2

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

I = 2/5 m × R2

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 × R2

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

Wrot = 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:

KErot = I × ω2/2

Power in rotational motion is

Prot = τ × ω

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

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