Motion in a Circle

Fig. 1.2 shows a cross-section through the cone and the steel ball.

On Fig. 1.2, draw labelled arrows to show the two forces acting on the ball.

Describe how the forces acting on the ball cause its acceleration to be centripetal.

The ball moves in a circle of radius $0.15\ \text{m}$.

Show that the speed of the ball is $1.4\ \text{m s}^{-1}$.

Calculate the angular speed $\omega$ of the ball.

$\omega$ = ______ $\text{rad s}^{-1}$

The speed of the ball is increased.

Explain why the radius of the circular path of the ball increases.

Define the radian.

Calculate the angular speed of the rear wheel.

angular speed = ______ $\text{rad s}^{-1}$

Calculate the period of rotation of the small cog.

period = ______ $\text{s}$

Show that the distance moved by point X on the chain during one full rotation of the small cog is $0.24\ \text{m}$.

Use the information in (b)(iii) to determine the angle through which the large cog rotates during one full rotation of the small cog.

angle = ______ $\text{rad}$

The chain of the bicycle in (b) is moved onto a smaller cog fixed to the rear wheel. The speed of the bicycle does not change.

Explain, without calculation, the effect of this change on the angular speed of the pedals.

Show that the radius of the circle around which Cambridge moves is $3.90 \times 10^{6}\text{ m}$.

Calculate the speed at which Cambridge moves around the circle.

$\text{speed}$ = ______ $\text{m s}^{-1}$

Determine the magnitude of the resultant force that acts to cause the circular motion of the student.

$\text{resultant force}$ = ______ $\text{N}$

On Fig. 1.2, draw an arrow to show the direction of the resultant force that acts on the student.

On Fig. 1.3, draw labelled arrows from the student to show the directions of the forces that act on the student to cause the resultant force in (b)(ii).

Define the radian.

an arrow, labelled V, showing the direction of the velocity of the modelling clay

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

Calculate the angular speed $\omega$ of the disc.

$\omega$ = ______ $\text{rad s}^{-1}$

Calculate the acceleration $a$ of the centre of gravity of the modelling clay.

$a$ = ______ $\text{m s}^{-2}$

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

Define the radian.

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.

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

$\omega$ = ______ $\text{rad s}^{-1}$

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

angle = ______ $\text{rad}$

Determine distance $L$.

$L$ = ______ $\text{m}$

Calculate the magnitude of the centripetal acceleration of the piece of modelling clay.

centripetal acceleration = ______ $\text{m s}^{-2}$

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.

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

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

Calculate $F$.

$F$ = ______ $\text{N}$

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

State what is meant by centripetal acceleration.

State what happens to the magnitude of the centripetal acceleration of the car as it moves around the loop from X to Y.

Explain, if the car remains in contact with the track, why the centripetal acceleration of the car at point Y must be greater than $9.8\text{ m s}^{-2}$.

The initial speed at which the car in (b) moves along the track is $3.8\text{ m s}^{-1}$.

Determine whether the car is in contact with the track at point Y. Show your working.

Suggest, with a reason but without calculation, whether your conclusion in (c) would be different for a car of mass $460\text{ g}$ moving with the same initial speed.

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

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

Calculate $F$.

$F$ = ______ $\text{N}$

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

Define the radian.

The Moon is approximately $3.8 \times 10^5\ \text{km}$ from Earth.
Estimate the minimum diameter of a circular crater on the Moon’s surface that can be seen using the telescope.

diameter = ______ $\text{km}$

Suggest why craters of the same diameter as that calculated in (i) but on the surface of Mars are not visible using this telescope.

By resolving the reaction force $R$ into two perpendicular components, show that the resultant force $F$ acting on the ball is given by the expression

State the significance of the force $F$ for the motion of the ball in the bowl.

The ball moves in a circular path of radius $14 \text{ cm}$. For this radius, the angle $\theta$ is $28^{\circ}$.

Calculate the speed of the ball.

speed = ______ $\text{m s}^{-1}$