CAIE A-Level Physics A2 12 Motion in a Circle Questions

Practise analysing circular motion with radians, angular and linear speed, centripetal acceleration and force, direction, units and assumptions.

Exam points

convert angular displacement and calculate angular speed, period and frequency in circular motion
relate angular and linear speed using v = rω
explain and calculate centripetal acceleration and force from radius, speed, angular speed and mass
identify the direction and physical source of centripetal force
check units and assumptions in uniform circular-motion problems

Covered syllabus points

12.1 Kinematics of uniform circular motion 12.2 Centripetal acceleration

[Maximum number: 9]

(a)

Define the radian.

[ 1 ]

(b)

A circular metal disc spins horizontally about a vertical axis, as shown in Fig. 1.1.

Fig. 1.1 (not to scale)

A piece of modelling clay is attached to the disc.
For the instant when the piece of modelling clay is in the position shown, draw on Fig. 1.1:

[ 1 ]

(i)

an arrow, labelled A , showing the direction of the acceleration of the modelling clay.

[ 1 ]

(c)

The metal disc in Fig. 1.1 has a radius of 9.3 cm .

The centre of gravity of the modelling clay is 1.2 cm from the rim of the disc and moves with a speed of $0.68 \mathrm{~ms}^{-1}$ .

[ 4 ]

(i)

Calculate the angular speed $\omega$ of the disc.
$\omega=$ $\mathrm{rad\,s^{-1}}$

[ 2 ]

$v=r \omega$ $\begin{aligned} \omega =0.68 /(0.093-0.012) =8.4 \mathrm{rad} \mathrm{~s}^{-1} \end{aligned}$

(ii)

Calculate the acceleration a of the centre of gravity of the modelling clay.
a= $\mathrm{ms}^{-2}$

[ 2 ]

$a=v^{2} / r$ or $a=r \omega^{2}$ $\begin{aligned} a =0.68^{2} /(0.093-0.012) \text { or }(0.093-0.012) \times 8.4^{2} =5.7 \mathrm{~m} \mathrm{~s}^{-2} \end{aligned}$

(d)

A second piece of modelling clay is attached to the disc in the position shown in Fig. 1.2.

Fig. 1.2

The second piece of modelling clay has a larger mass than the first piece.
By placing one tick $(\checkmark)$ in each row, complete Table 1.1 to show how the quantities indicated compare for the two pieces of modelling clay.

Table 1.1

[ 3 ]

[Maximum number: 7]

(a)

With reference to velocity and acceleration, describe uniform circular motion.

[ 2 ]

(b)

Two cars are moving around a horizontal circular track. One car follows path X and the other follows path Y , as shown in Fig. 1.1.

Fig. 1.1 (not to scale)

The radius of path X is 318 m . Path Y is parallel to, and 27 m outside, path X . Both cars have mass 790 kg . The maximum lateral (sideways) friction force F that the cars can experience without sliding is the same for both cars.

[ 5 ]

(i)

The maximum speed at which the car on path X can move around the track without sliding is $94 \mathrm{~m} \mathrm{~s}^{-1}$ .

Calculate F.
F= N

[ 2 ]

$F=m v^{2} / r$

or
$v=r \omega \text { and } F=m r \omega^{2}$ $\begin{aligned} F =790 \times 94^{2} / 318 =22000 \mathrm{~N} \end{aligned}$

(ii)

Both cars move around the track. Each car has the maximum speed at which it can move without sliding.

Complete Table 1.1, by placing one tick in each row, to indicate how the quantities indicated for the car on path Y compare with the car on path X .

Table 1.1

[ 3 ]

[Maximum number: 7]

A steel sphere of mass 0.29 kg is suspended in equilibrium from a vertical spring. The centre of the sphere is 8.5 cm from the top of the spring, as shown in Fig. 2.1.

Fig. 2.1

The sphere is now set in motion so that it is moving in a horizontal circle at constant speed, as shown in Fig. 2.2.

Fig. 2.2

The distance from the centre of the sphere to the top of the spring is now 10.8 cm .

(a)

Explain, with reference to the forces acting on the sphere, why the length of the spring in Fig. 2.2 is greater than in Fig. 2.1.

[ 3 ]

(b)

Use the information in (b) to determine the centripetal acceleration of the sphere.

\text{centripetal acceleration = ............................................... }\mathrm{m}\,\mathrm{s}^{-2}

[ 1 ]

centripetal acceleration $=(T \sin \theta) / m$ $=\left(3.2 \times \sin 27^{\circ}\right) / 0.29$ $=5.0 \mathrm{~m} \mathrm{~s}^{-2}$

(c)

Calculate the period of the circular motion of the sphere.
period =

[ 3 ]

$a=r \omega^{2} \text { and } \omega=2 \pi / T$

or
$a=v^{2} / r \text { and } v=2 \pi r / T$ $\begin{aligned} T =2 \pi \times \sqrt{ }(0.049 / 5.0) =0.62 \mathrm{~s} \end{aligned}$

[Maximum number: 10]

(a)

Define the radian.

[ 1 ]

(b)

The minute hand of a clock revolves at constant angular speed around the face of the clock, completing one revolution every hour. A small piece of modelling clay is attached to the hand with its centre of gravity at a distance L from the fixed end of the hand, as shown in Fig. 1.1.

direction of revolution of minute hand

Calculate the angular speed $\omega$ of the minute hand.

[ 2 ]

$\omega=2 \pi / T$ $\begin{aligned} =2 \pi /(1.0 \times 60 \times 60) =1.7 \times 10^{-3} \mathrm{rad} \mathrm{~s}^{-1} \end{aligned}$

(c)

During a time interval of 1400 s , the centre of gravity of the piece of modelling clay in Fig. 1.1 moves through a total distance of 0.44 m .

[ 5 ]

(i)

Calculate the angle through which the minute hand moves in this time interval.

angle =rad [1]

[ 1 ]

$\begin{aligned} \text { angle } =1.7 \times 10^{-3} \times 1400 =2.4 \mathrm{rad} \end{aligned}$

(ii)

Determine distance L.

L =m [2]

[ 2 ]

$\begin{aligned} L & =\text { arc length } / \text { angle } & =0.44 / 2.4 \end{aligned}$

or
$L=0.44 \times(3600 / 1400) / 2 \pi$ $L=0.18 \mathrm{~m}$

(iii)

Calculate the magnitude of the centripetal acceleration of the piece of modelling clay.
centripetal acceleration = $\mathrm{ms}^{-2}$

[ 2 ]

$a=r \omega^{2}$ $\begin{aligned} =0.18 \times\left(1.745 \times 10^{-3}\right)^{2} =5.5 \times 10^{-7} \mathrm{~m} \mathrm{~s}^{-2} \end{aligned}$

(d)

Use your answer in (c)(iii) to explain why the variation with time of the magnitude of the force exerted by the minute hand on the piece of modelling clay is negligible as the minute hand undergoes one full revolution.

[ 2 ]

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Covered syllabus points

Question 1

Question (a)

Question (b)

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Question (c)

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Question (ii)

Question (d)

Question 2

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Question (ii)

Question 3

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Question (c)

Question 4

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Question (ii)

Question (iii)

Question (d)