EduNinja

Connect the descriptions

For uniform circular motion,

$v=\frac{2\pi r}{T}=\omega r$

Angular speed $\omega$ is the same for all points on a rigid rotating body, while linear speed increases with distance from the axis.

Use the period

One revolution takes period $T$ , so $\omega=2\pi/T$ . Keep radians and seconds consistent.

Compare points on one disk

If one point is twice as far from the centre, its linear speed is twice as large at the same angular speed; its centripetal acceleration is also twice as large.

Common trap

Do not assume equal linear speeds for all points on a rotating disk. Equal angular speed does not mean equal tangential speed.

[ 1 ]

A disk of radius R rotates about its axis with angular speed $\omega$ . Point X is at a distance of $\frac{R}{2}$ from the centre and point Y is on the circumference.

What are the ratios of the linear speeds and the centripetal acceleration of X to Y.
The linear speed of X is $v_{X}$ and its acceleration is $a_{X}$ ; the linear speed of Y is $v_{Y}$ and its acceleration is $a_{Y}$ .

Linear speeds $\frac{\boldsymbol{v}_{\mathbf{X}}}{\boldsymbol{v}_{\mathbf{Y}}}$

Acceleration $\frac{\mathbf{a}_{\mathbf{X}}}{\mathbf{a}_{\mathbf{Y}}}$

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{1}{2}$

1

$\frac{1}{4}$

1

$\frac{1}{2}$

Build the force model

Choose the system, draw a labelled free-body diagram, classify the interactions and resolve components. Apply Newton’s laws with the correct boundary: contact forces, field forces, friction, tension, buoyancy and restoring forces each have their own direction and conditions.

Track momentum

Use $ec p=m ec v$ , $ec J=\Delta ec p$ and momentum conservation only after checking external impulse. Distinguish elastic and inelastic collisions, explosions and energy transfer.

Track circular motion

The inward resultant provides $a_c=v^2/r=\omega^2r$ . It may come from tension, gravity, normal, friction or a field force. Angular and linear descriptions are linked by $v=\omega r=2\pi r/T$ .

Final checks

Ask: Which body is the system? Which forces are external? Is mass constant? Is acceleration uniform or radial? Is kinetic energy conserved, transferred or increased?

A.2.26—Angular and linear speed

Link Angular and Linear Speed

Retrieve the A.2 Forces and Momentum Model

A.2.26 Exam Analysis