Circular Motion & Radians | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

What circular motion is

means motion along a circular path.
The centre is the fixed point in the middle of the circle.
The radius r is the straight-line distance from the centre to the moving object. This distance stays constant.
The object may travel around a full circle or only part of a circle. Its direction changes as it moves.
Examples include a point on a turning wheel, a stopper on a string and a satellite in a circular orbit.

Circular motion is motion along a circular path. The radius is the distance from the centre to the object.

Your turnQuick check

An object moves on a circular path of radius 0.40 m. What distance stays constant? Does the direction of motion stay constant?

Show worked answer

The distance from the object to the centre stays at 0.40 m. The direction of motion changes.

Angle turned around the centre

We use an angle to say how far around the circular path the object has moved.
tells us how far an object has turned around the centre. Its symbol is $\theta$ .
The angle is measured from a chosen starting line. A quarter turn is 90°. A half turn is 180°. A full turn is 360°.
Physics normally measures this angle in .

When the arc length s equals the radius r, the angle at the centre is one radian.

Definition

1 mark

What is meant by one radian?

Model answer: The angle at the centre of a circle when the arc length is equal to the radius (9702/42/M/J/25 Q1(a)).

\theta = \dfrac{s}{r}

Symbols

$\theta$ = angular displacement (rad)
$s$ = arc length (m)
$r$ = radius of the circle (m)

The is the distance measured along the circle.
The metre units in $s/r$ cancel, so no unit is left. In an answer, still write rad to show that the angle is in radians.
A full circle has arc length $2\pi r$ , so a full turn is $2\pi$ rad.

Degrees and radians

A full turn is both $360^\circ$ and $2\pi\text{ rad}$ .
Multiply degrees by $\pi/180$ to get radians.
Multiply radians by $180/\pi$ to get degrees.

Worked example

Changing degrees to radians

Write 60° in radians.

Multiply by $\pi/180$ .
$\theta=60\times\dfrac{\pi}{180}=\dfrac{\pi}{3}$
Write the decimal value to three significant figures.
$\theta=1.05\text{ rad}$

Answer

\theta=\pi/3\approx1.05\text{ rad}

Your turnQuick check

Convert 150° to radians. Then convert 2.40 rad to degrees.

Show worked answer

150\times\dfrac{\pi}{180}=2.62\text{ rad}

2.40\times\dfrac{180}{\pi}=138^\circ

Using arc length

Use the angular-displacement formula when the question gives a distance around a circle and a radius.
Rearrange the same formula when the question asks for arc length.
The angle in these equations must be in radians.

Worked example

Finding an angle from an arc

A chain moves 0.24 m around a gear of radius 0.15 m. Find the angle turned by the gear (9702/42/M/J/25 Q1(b)(iv)).

Substitute the arc length and radius.
$\theta=\dfrac{s}{r}=\dfrac{0.24}{0.15}=1.6\text{ rad}$

Answer

\theta=1.6\text{ rad}

Your turnQuick check

A point turns through 1.2 rad on a circle of radius 0.40 m. Find the arc length.

Show worked answer

Rearrange for arc length, then substitute the values.

s=r\theta=0.40\times1.2=0.48\text{ m}

Common mistake

Use radian mode when an angle in a trigonometric equation is written in radians. Use degree mode for an angle marked with the degree sign.

What circular motion is

means motion along a circular path.
The centre is the fixed point in the middle of the circle.
The radius r is the straight-line distance from the centre to the moving object. This distance stays constant.
The object may travel around a full circle or only part of a circle. Its direction changes as it moves.
Examples include a point on a turning wheel, a stopper on a string and a satellite in a circular orbit.

Circular motion is motion along a circular path. The radius is the distance from the centre to the object.

Your turnQuick check

An object moves on a circular path of radius 0.40 m. What distance stays constant? Does the direction of motion stay constant?

Show worked answer

The distance from the object to the centre stays at 0.40 m. The direction of motion changes.

Angle turned around the centre

We use an angle to say how far around the circular path the object has moved.
tells us how far an object has turned around the centre. Its symbol is $\theta$ .
The angle is measured from a chosen starting line. A quarter turn is 90°. A half turn is 180°. A full turn is 360°.
Physics normally measures this angle in .

When the arc length s equals the radius r, the angle at the centre is one radian.

Definition

1 mark

What is meant by one radian?

Model answer: The angle at the centre of a circle when the arc length is equal to the radius (9702/42/M/J/25 Q1(a)).

\theta = \dfrac{s}{r}

Symbols

$\theta$ = angular displacement (rad)
$s$ = arc length (m)
$r$ = radius of the circle (m)

The is the distance measured along the circle.
The metre units in $s/r$ cancel, so no unit is left. In an answer, still write rad to show that the angle is in radians.
A full circle has arc length $2\pi r$ , so a full turn is $2\pi$ rad.

Degrees and radians

A full turn is both $360^\circ$ and $2\pi\text{ rad}$ .
Multiply degrees by $\pi/180$ to get radians.
Multiply radians by $180/\pi$ to get degrees.

Worked example

Changing degrees to radians

Write 60° in radians.

Multiply by $\pi/180$ .
$\theta=60\times\dfrac{\pi}{180}=\dfrac{\pi}{3}$
Write the decimal value to three significant figures.
$\theta=1.05\text{ rad}$

Answer

\theta=\pi/3\approx1.05\text{ rad}

Your turnQuick check

Convert 150° to radians. Then convert 2.40 rad to degrees.

Show worked answer

150\times\dfrac{\pi}{180}=2.62\text{ rad}

2.40\times\dfrac{180}{\pi}=138^\circ

Using arc length

Use the angular-displacement formula when the question gives a distance around a circle and a radius.
Rearrange the same formula when the question asks for arc length.
The angle in these equations must be in radians.

Worked example

Finding an angle from an arc

A chain moves 0.24 m around a gear of radius 0.15 m. Find the angle turned by the gear (9702/42/M/J/25 Q1(b)(iv)).

Substitute the arc length and radius.
$\theta=\dfrac{s}{r}=\dfrac{0.24}{0.15}=1.6\text{ rad}$

Answer

\theta=1.6\text{ rad}

Your turnQuick check

A point turns through 1.2 rad on a circle of radius 0.40 m. Find the arc length.

Show worked answer

Rearrange for arc length, then substitute the values.

s=r\theta=0.40\times1.2=0.48\text{ m}

Common mistake

Use radian mode when an angle in a trigonometric equation is written in radians. Use degree mode for an angle marked with the degree sign.