Linear Speed, Wheels & Gears | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

Speed along the circle

is the speed along the circular path. Its symbol is v and its unit is m s⁻¹.
A point farther from the centre travels a longer arc in the same time, so it has a larger linear speed when the angular speed is fixed.

v=r\omega

Symbols

$v$ = speed along the circle (m s⁻¹)
$r$ = radius of the path (m)
$\omega$ = angular speed (rad s⁻¹)

The object turns through an angle during time $t$ . The distance travelled is the arc length for that angle.

\theta=\omega t

s=r\theta=r\omega t

v=\dfrac{s}{t}=r\omega

Worked example

A point on a wheel

A point is 0.20 m from the centre of a wheel. The wheel turns at 3.0 rad s⁻¹. Find the speed of the point.

Substitute the radius and angular speed.
$v=r\omega$
$v=0.20\times3.0$
$v=0.60\text{ m s}^{-1}$

Answer

v=0.60\text{ m s}^{-1}

First decide what is joined. One solid wheel has the same angular speed. One chain has the same linear speed at both gear edges.

System	Quantity that is the same	What changes
points on one solid wheel	angular speed	a larger radius gives a larger speed v
gear edges joined by one chain	linear speed v	a smaller radius gives a larger angular speed

Side view of a bicycle. Labels point to the small rear gear, the large front gear and the chain that has the same linear speed at both gears. — The chain joins the large front gear to the small rear gear. The small rear gear turns with the rear wheel.

Your turn9702/42/M/J/25 Q1(c)

A bicycle keeps the same road speed. Its chain is moved to a smaller rear gear. Explain why the pedals turn more slowly.

Show worked answer

The rear wheel keeps the same angular speed. A smaller rear gear has a smaller edge speed. The chain speed therefore becomes smaller. The front gear keeps the same radius, so its angular speed, and the angular speed of the pedals, also become smaller.

Worked example

The rear wheel of a bicycle

A bicycle moves at 17 m s⁻¹. Its rear wheel has radius 0.46 m and rolls without sliding. Find the angular speed of the wheel (9702/42/M/J/25 Q1(b)(i)).

Rearrange the speed equation for angular speed.
$\omega=\dfrac{v}{r}$
Substitute the speed and radius.
$\omega=\dfrac{17}{0.46}=37\text{ rad s}^{-1}$

Answer

\omega=37\text{ rad s}^{-1}

Your turnQuick check

Two points are 0.20 m and 0.50 m from the centre of one solid wheel. The wheel turns at 4.0 rad s⁻¹. Find both speeds.

Show worked answer

Both points have the same angular speed.

v_1=0.20\times4.0=0.80\text{ m s}^{-1}

v_2=0.50\times4.0=2.0\text{ m s}^{-1}

Common mistake

Hertz means turns per second. Radians per second means angle turned per second. The two units describe different quantities.

Speed along the circle

is the speed along the circular path. Its symbol is v and its unit is m s⁻¹.
A point farther from the centre travels a longer arc in the same time, so it has a larger linear speed when the angular speed is fixed.

v=r\omega

Symbols

$v$ = speed along the circle (m s⁻¹)
$r$ = radius of the path (m)
$\omega$ = angular speed (rad s⁻¹)

The object turns through an angle during time $t$ . The distance travelled is the arc length for that angle.

\theta=\omega t

s=r\theta=r\omega t

v=\dfrac{s}{t}=r\omega

Worked example

A point on a wheel

A point is 0.20 m from the centre of a wheel. The wheel turns at 3.0 rad s⁻¹. Find the speed of the point.

Substitute the radius and angular speed.
$v=r\omega$
$v=0.20\times3.0$
$v=0.60\text{ m s}^{-1}$

Answer

v=0.60\text{ m s}^{-1}

First decide what is joined. One solid wheel has the same angular speed. One chain has the same linear speed at both gear edges.

System	Quantity that is the same	What changes
points on one solid wheel	angular speed	a larger radius gives a larger speed v
gear edges joined by one chain	linear speed v	a smaller radius gives a larger angular speed

Your turn9702/42/M/J/25 Q1(c)

A bicycle keeps the same road speed. Its chain is moved to a smaller rear gear. Explain why the pedals turn more slowly.

Show worked answer

Worked example

The rear wheel of a bicycle

A bicycle moves at 17 m s⁻¹. Its rear wheel has radius 0.46 m and rolls without sliding. Find the angular speed of the wheel (9702/42/M/J/25 Q1(b)(i)).

Rearrange the speed equation for angular speed.
$\omega=\dfrac{v}{r}$
Substitute the speed and radius.
$\omega=\dfrac{17}{0.46}=37\text{ rad s}^{-1}$

Answer

\omega=37\text{ rad s}^{-1}

Your turnQuick check

Two points are 0.20 m and 0.50 m from the centre of one solid wheel. The wheel turns at 4.0 rad s⁻¹. Find both speeds.

Show worked answer

Both points have the same angular speed.

v_1=0.20\times4.0=0.80\text{ m s}^{-1}

v_2=0.50\times4.0=2.0\text{ m s}^{-1}

Common mistake

Hertz means turns per second. Radians per second means angle turned per second. The two units describe different quantities.