P.

AL Mechanics Circular Motion

Name________________
Class____________( )

Date Grade Correction

1
2
3
Your Own Notes:
 P.2

Your Own Notes:

 P.3

Circular Motion HW1

Grade:___________
Correction:_____________________________
1. A boy is running on a circular path with an average angular velocity of 0.5 rad s−1. What is the period of
his motion?
A. 2 s
B. 3.14 s
C. 6.28 s
D. 12.6 s
Answer: D

2. Which of the following objects has the highest average speed?

A. A man runs at 4 m s−1 on a straight road.
B. A man runs at 0.4 rad s−1 on a circular path of a radius of 10 m.
C. A woman runs on a circular path of radius 10 m with a period of 10 s.
D. A woman runs on a straight path of 50 m in 10 s.
Answer: C

3. Which of the following objects has the largest magnitude of acceleration?

A. A car of mass 1500 kg travels with a uniform speed of 10 m s−1 on a straight
road.
B. A car of mass 1500 kg travels with a uniform speed of 10 m s−1 on a circular
track of a radius of 50 m.
C. A van of mass 5000 kg speeds up from rest to 10 m s−1 in 5 s.
D. A van of mass 5000 kg travels with a uniform speed on a circular track of a
radius of 50 m with a period of 20 s.
Answer: D

4. A car moves through a circular arc of length l and radius r with a constant speed in time t. What is its
angular velocity?
 P.4

A.

B.

C.

D.
Answer: C

5. An object is in uniform circular motion. Which of its quantities will increase if its mass and the radius of
the path are doubled while its speed remains unchanged?
(1) centripetal acceleration
(2) centripetal force
(3) angular speed
A. (1) only
B. (2) only
C. (3) only
D. none of the above
Answer: D

6. An toy car of mass 2.5 kg moves on a horizontal circular path with a constant speed. The radius of the
path is 2 m and it takes 12 s for the car to complete one revolution. What is the magnitude of the centripetal
force acting on the car?
A. 0.524 N
B. 0.548 N
C. 1.31 N
D. 1.37 N
Answer: D

7. An object of mass 5 kg moves with a constant speed along a horizontal circular path of radius 2 m. It
takes 4 s to complete one revolution. What is the centripetal force acting on the object?

A.

B.
 P.5

C.
D.
Answer: C

8. A ball B is swung on a horizontal plane as shown.

Which of the following diagram best illustrates the acceleration a of the ball?
A. no acceleration B.

C. D.
Answer: C

9. A particle is moving on a horizontal circular path with a constant angular velocity. Which of the following
quantities of the particle remain constant?
(1) linear speed
(2) kinetic energy
(3) linear momentum
A. (1) and (2) only
B. (1) and (3) only
C. (2) and (3) only
D. (1), (2) and (3)
Answer: A
 P.6

10. The minute hand of a clock is 20 cm long.

Photo Credit: Erashov | Shutterstock.com

What is its angular speed?

A.

B.

C.

D.

Answer: B

11. The Moon orbits around the Earth in 27.3 days and its average distance from the Earth is 3.84 × 108 m.

Photo Credit: FrameAngel | Shutterstock.com

Assuming that the Moon is moving in a circular orbit, what is its angular speed?
A. 5.99 × 10−9 rad s−1
B. 1.33 × 10−6 rad s−1
C. 2.66 × 10−6 rad s−1
 P.7

D. 0.230 rad s−1

Answer: C

12. A disc of diameter 20 cm completes 100 revolutions in 3 minutes. What is the period of the rotation?
A. 0.36 s
B. 0.555 s
C. 1.8 s
D. 9 s
Answer: C

13. The angular speed of Mercury is about 8.267 × 10−7 rad s−1 when it orbits around the Sun.

Photo Credit: NASA/JPL

Assuming that its orbit is circular with a radius of 5.791 × 1010 m, what is the linear speed of Mercury?
A. 4.79 × 104 m s−1
B. 1.50 × 105 m s−1
C. 3.01 × 105 m s−1
D. 6.02 × 105 m s−1
Answer: A
 P.8

14.

Students A and B are travelling on two concentric circular paths as shown. If they have the same angular
velocity, find the ratio of the linear velocity of A to that of B.
A. 1 : 2
B. 1 : 3
C. 1 : 4
D. 1 : 9
Answer: B
 P.9

15. A bead A of mass 2 kg is connected to a fixed point O with an inextensible string of 1.5 m long. The
bead is whirled around O as shown and the maximum tension that can be withstood by the string is 30 N.

What is the maximum speed of the bead before the string breaks?
A. 4.74 m s−1
B. 5.81 m s−1
C. 6.71 m s−1
D. 22.5 m s−1
Answer: A

16. A ball is hanged by an inextensible string and move on a horizontal circle. Which of the following
diagrams best illustrates the resultant force F acting on the ball?
 P.10

A. B.

C. D.

Answer: B

17. In an experiment, a rubber bung is connected to a weight via a glass tube by an inextensible string. The
rubber bung is then whirled around horizontally at a steady speed v. The mass of the rubber bung and the
weight are M and m respectively. The length of the string extended from the glass tube to the rubber bung is
L. The angle between the string and the vertical is θ.

glass tube

rubber bung

weight

What is the tension in the string?

A.

B.
 P.11

C.

D.
Answer: D

18. In the conical pendulum as shown, the bead is whirled on a horizontal circle. The string makes an angle
of θ with the vertical and its tension is T.

If the bead is whirled at a higher speed,

(1) θ increases.
(2) T increases.
(3) the radius of the circular path increases.
A. (1) and (2) only
B. (1) and (3) only
C. (2) and (3) only
D. (1), (2) and (3)
Answer: D

19. A car travels along the path PQRST as shown with a uniform speed.
 P.12

In which path segment does the car have the largest acceleration?
A. PQ
B. QR
C. RS
D. ST
Answer: B

Circular Motion HW2

Grade:___________
Correction:_____________________________

1. A wheel has a radius of 0.3 m and is rotating at 600 revolutions per minute.
(a) What is the period of the motion of the wheel in seconds?
(2 marks)
(b) What is the angular speed of the wheel?
(2 marks)
(c) What is the linear speed of a point on the edge of the wheel?
(2 marks)
Answer:

(a) The period is . (1M+1A)

(b) The angular speed is . (1M+1A)

 P.13

(c) The linear speed is rω = 0.3 × 62.83 ≈ 18.8 m s−1. (1M+1A)

2. A potter’s wheel rotates steadily at 180 revolutions per minute (rpm).

Photo Credit: Imagemore Co., Ltd

(a) What is the frequency of rotation (in Hz)?

(2 marks)
(b) What is the centripetal acceleration of a point 10 cm from the rotation axis?
(2 marks)
Answer:

(a) The frequency is . (1M+1A)

(b) The centripetal acceleration is . (1M+1A)

3. A powerboat of total mass 3800 kg makes a sharp turn with a uniform speed of
20 m s−1. The path describes an arc of a circle of a radius of 24 m.

Photo Credit: EPSTOCK | Fotolia.com

(a) What is the centripetal force provided to the powerboat during its turn?
(2 marks)
(b) With the force in (a) unchanged, would the boat trace out an arc of a larger or
smaller radius if the speed becomes higher?
 P.14

(1 mark)
Answer:

(a) The centripetal force is . (1M+1A)

(b) The powerboat will trace out an arc of a larger radius. (1A)

4. A puck of mass 0.5 kg, attaching to one end of a string, is moving with a uniform speed in a horizontal
circle of a radius of 1.5 m on a smooth table. The string breaks if the tension T exceeds 100 N.

(a) What is the maximum value of v before the string breaks? What is the
corresponding angular speed ω?
(4 marks)
(b) The string suddenly breaks. Describe the subsequent motion of the puck.
(2 marks)

(a) Applying , the maximum value, vmax is

(1M)

The maximum value of v is 17.3 m s−1. (1A)

The angular speed ω is . (1M+1A)

(b) The puck moves uniformly (1A) along the direction just before the string breaks
(tangential to the circular path at the point of breaking) (1A).
5. A roller coaster car of total mass 320 kg travels around a horizontal circular track of a radius of 32 m with
a uniform speed of 16 m s−1.
(a) What is the magnitude and direction of the resultant force acting on the car by
the track?
(4 marks)
(b) Hence, or otherwise, determine the angle the track should be banked such that no
frictional force is needed by the car to travel around the track.
(1 mark)
Answer:
(a) The horizontal component of the force acting on the car by the track provides the

centripetal force, . (1M)

The vertical component of the force acting on the car by the track balances the
 P.15

weight of the car, mg = 320 × 10 = 3200 N. (1M)

Therefore the magnitude of the resultant force is

. (1A)

The force points to above the horizontal direction. (1A)

(b) The angle is 51.3° above the horizontal direction. (1A)

6. A car moving at a uniform speed of 10 m s−1 turns around a corner, which is an arc of a circle of a radius
of 25 m. The maximum frictional force developed between the car tyre and the road surface is 0.7 times the
weight of the car.
(a) Will the car skid as it turns around the corner?
(3 marks)
(b) What is the maximum speed vmax of the car before it skids?
(1 mark)
Answer:
(a) Let m kg be the mass of the car.

The centripetal force needed is . (1M)

The maximum frictional force is 0.7mg = 0.7 × m × 10 = 7m. (1M)

Therefore the car will not skid. (1A)

(b) The speed vmax can be determined by

(1A)

7. An aeroplane of mass 12 000 kg is flying at a constant speed of 90 m s−1 in a horizontal circle of a radius
of 2400 m. The lift force U is perpendicular to the wings.

(a) Find the centripetal force needed by the aeroplane.

(2 marks)
(b) At what angle to the vertical, θ must the wings be banked?
 P.16

(3 marks)
Answer:

(a) Applying , the centripetal force needed is

. (1M+1A)

(b) The horizontal component of U provides the centripetal force. (1M)

The vertical component of U balances the weight. (1M)

Consider the horizontal and vertical components of U.

(1A)

8. A stone of mass m = 50 g is whirled in a horizontal circle at a steady speed as shown. The length of the
string is l = 0.8 m and makes an angle θ with the vertical.
 P.17

stone

(a) Suppose θ = 60°. Find the angular speed of the stone.

(3 marks)
(b) If the experiment is carried out on the surface of the Moon where the
acceleration due to gravity is smaller, should the stone be whirled faster or
slower to keep the same angle θ? Briefly explain your answer.
(2 marks)
Answer:
(a) Let T be the tension of the string.
The vertical component of T is equal to the weight of the stone, i.e.

(1M)

The horizontal component of T provides the centripetal force, i.e.

(1M+1A)

(b) From (a), we have

(1A)

Therefore, where g becomes smaller, the stone have to be whirled slower to keep the same angle θ. (1A)
 P.18

9. In an experiment, a rubber bung is connected to a weight via a glass tube by an inextensible string. The
rubber bung is then whirled around horizontally at a steady angular speed ω. The mass of the rubber bung
and the weight are M and m respectively. The length of the string extended from the glass tube to the rubber
bung is L. The angle between the string and the vertical is θ.

glass tube

rubber bung

weight

(a) Derive an expression to describe the relation between M, m, ω and L.

(3 marks)
(b) The bung completes 20 revolutions in 12 s.
(i) Find ω.
(2 marks)
(ii) Suppose L = 1.2 m. Find θ.
(2 marks)
Answer:
(a) The tension in the string T is equal to mg.
The horizontal component of the tension, i.e. mg sin θ provides the centripetal
force. (1M)

The centripetal force provided is Mω2L sin θ. (1M)

Therefore,

(1A)

(b) (i) (1M+1A)

(ii) Consider the horizontal and vertical components of the tension in the string.

(1M+1A)
 P.19

10. A car of mass 1500 kg moving at a uniform speed of 10 m s−1 turns around a corner, which is an arc of a
circle of a radius of 50 m.

(a) Suppose the road is level such that the centripetal force is solely provided by the
frictional force f acting on the car by the road surface. Find f.
(2 marks)
(b) Suppose the road is banked such that the centripetal force is solely provided by
the normal reaction R of the car. Find the angle θ at which the road is banked.
(3 marks)
Answer:

(a) Applying , . (1M+1A)

(b) The normal reaction R is equal to component of the weight perpendicular to the
road surface, i.e.

(1) (1M)

In this case, the horizontal component of the normal reaction is equal to 3000 N,
i.e.

(2) (1M)

Substitute (1) into (2),

(1A)
 P.20

Circular Motion HW3

Grade:___________
Correction:_____________________________

1. A toy aeroplane of mass 0.5 kg flies in a horizontal circle. The lifting force U = 10 N is perpendicular to
the wings. The aeroplane completes one revolution in 6 s.

(a) Find the angular speed of the aeroplane.

(2 marks)
(b) Find θ, the angle between the vertical and the lifting force.
(2 marks)
(c) Find the radius of the circle r.
(2 marks)
(d) The lifting force U remains unchanged. How does θ change if
(i) the mass of the aeroplane becomes smaller.
(ii) the aeroplane travels at a higher speed but the radius of the circle remains
unchanged.
(2 marks)
Answer:

(a) The angular speed is . (1M+1A)

(b) The vertical component of the lifting force balances the weight of the aeroplane,
i.e.

(1M+1A)

(c) The horizontal component of the lifting force provides the centripetal force, i.e.
 P.21

(1M+1A)

(d) (i) increases (1A)

(ii) increases (1A)

2. A ball is whirled in a horizontal circle of a radius of 0.4 m on a plane 1.2 m above the ground. It is
suddenly released and it lands on a position 2 m away as shown.

(a) What is the centripetal acceleration of the ball just before it is released?
(4 marks)
(b) Find the angle between the string and vertical just before the ball is released.
(3 marks)
(c) How do (1) the time of flight in the air and (2) the horizontal distance travelled
by the ball change in the following cases.
(i) using a heavier ball without changing the angular speed
(ii) increasing the whirling speed without changing the initial height of the ball
from the ground
(2 marks)
Answer:
(a) Let t be the time of flight of the ball and u be the horizontal speed of the ball.
Consider the horizontal motion of the ball.
ut = 2 (1M)

Consider the vertical motion of the ball.

 P.22

Thus u = 4.082 m s−1. (1M)

Applying , the centripetal acceleration is

(1M+1A)

(b) Let m be the mass of the ball.

The vertical component of the tension in the string balances the weight of the
ball, i.e. mg = 10m. (1M)

The horizontal component of the tension provides the centripetal force, i.e.
41.67m. (1M)

The angle is . (1A)

(c) (i) (1) unchanged

(2) unchanged (1A)

(ii) (1) unchanged

(2) increases (1A)

3. There is an amusement ride called the Rotor. It consists of a large barrel rotating at about 33 revolutions
per minute (rpm). When the barrel attains its full speed, the riders have already been pinned against the
wall. Consider the barrel below. A boy of mass 45 kg has already been pinned against the wall.

(a) Sketch a free-body diagram of the boy.

(2 marks)
(b) The centripetal acceleration is about 1.5g, i.e. 15 m s−2. Find r.
(2 marks)
(c) What is the linear speed of the boy?
 P.23

(2 marks)
(d) While the boy is still pinned against the wall, the barrel rotates uniformly at a lower speed. The
resultant force acting on the boy now becomes 450 N. Find the linear speed of the boy again.
(3 marks)
Answer:
(a)

f = friction
N = normal reaction from the wall
W = weight of the boy

(1A for friction, normal reaction and weight, 1A for correct directions)

(b) The angular speed is .

Applying a = rω2, we have

(1M+1A)

(c) The linear speed is v = rω = 4.340 589 × 1.1π = 15 m s−1. (1M+1A)

(d) Applying F = ma, we have

(1M)

Applying , we have

(1M+1A)