Lesson 11: Angular Motion

Angular Terms

Angular Displacement

Defined as the angle and the direction through which a body turns

Measured using rad or revolution

Denoted by theta

Important note: 1 revolution (1 turn) = 360 degrees = 2pi radians

Angle in radians

Ratio of two lengths

Pure number, which means there is no dimensions

image

Angular Velocity

Defined as the rate of change of angular displacement with respect to time

Angular velocity (Mathematical form):

image

Measured using radians per second [rad/s]

Can also be represented as revolution per minute [rpm]

Important thing to note: 1 rpm = 2pi/60 rad/s

Angular Acceleration

Defined as the rate of change of angular velocity with respect to time

Angular Acceleration (Mathematical form):

image

Measured using radians per second squared [rad/s2]

Uniform Motion

A motion when the angular acceleration is constant

Three equations of rotational motion with constant angular acceleration are:

image

Important Formulas

Comparison of Linear and Rotational Motion Equations (Uniform Motion) with Constant Acceleration:

image

RELATIONSHIP between LINEAR and ANGULAR Motion:

image

Normal or Centripetal Acceleration

For a body moving in a circular motion

Have an acceleration an, that is radially inwards towards the centre of the circle with a magnitude given by

image

Important Note: To find the total acceleration of acceleration and tangential acceleration:

image

image

image

Translational Motion & Rotational Motion

Translational Motion

For a rolling ball,

Every particle in the rigid body has the same instantaneous velcocity

Means that there is no rotation

image

Rotational Motion

For a rolling ball,

Every particle in the rigid body has the same angular velocity

Also travels in circles around a fixed axis

image

Rolling Motion

Combination of Translational and Rotational Motion

image

Translational Motion vs Rotational Motion

Rotational Motion: Motion about a fixed axis, all the particles have the same instantaneous angular velocity and travel in circles about the axis of rotation

General Rigid-body motion: A combination of TRANSLATIONAL and ROTATIONAL motion

Common example: Rolling

Combination of both TRANSLATIONAL and ROTATIONAL motion of a rigid body

Rolling without SLIPPING is a combination of ROTATION and TRANSLATION

Point of contact is instantaneously at rest, point 3 at the top of the wheels moving forward twice as fast as the center of mass; points 2 and 4 at the side have velocities at 45 degrees to the horizontal.

image