7 Motion in a Circle
8
Q1. Convert the following angles into radians. (Sang W18)
(a) 30°
(b) 210°
(c) 0.05°
Q2. Convert the following angles from radians into degrees. (Sang W18)
(a) 1.0 rad
(b) 4.0 rad
(c) 0.15 rad
Q3. Change the following angles into radians. Do not use a calculator. You can leave p in any of your
answers. (Hutchings pg. 109)
(a) two revolutions
(b) 135°
(c) five right-angles
(d) 50 revolutions
(e) 10°
(f) the angle the minute hand of a watch rotates through in a day
Q4. The planet Mercury takes 88 days to orbit once round the Sun. Calculate its angular displacement
in radians during a time interval of:
(a) 44 days,
(b) 1 day.
(Sang W18)
Q5. The laser on a CD player is 5 cm from the centre hole of the disc. What length of the disc is scanned
by the laser when the disc turns through an angle of 0.45 radians?
Q6. A clock has a second hand that is 5.0 cm long, a minute hand that is 5.0 cm long and an hour hand
that is 4.0 cm long. Find the angular velocity of each of the hands and the linear speed of the tip of each
hand. Assume that there is no jerkiness in their movement. (Hutchings pg. 110)
Q7. Give the angular velocity of each of the following.
a. the Earth in its orbit around the Sun,
b. the Earth about its axis,
c. the Moon in its orbit around the Earth, and
d. the Moon about its axis. (Hutchings pg. 110)
Q8. A car is travelling along a circular path with linear speed 15 m s-1 and angular speed 0.36 rad s-1.
What is the radius of curvature of the track? (Mee pg. 185)
Q9. Find the acceleration of the Moon. It travels at a constant speed of 1020 m s-1 and takes 27.3 days
for a complete revolution of the Earth. (Hutchings pg. 111)
Q10. In each case below, state what provides the centripetal force on the object.
(a) A car travels at a high speed round a sharp corner.
(b) A planet orbits the Sun.
(c) An electron orbits the positive nucleus of an atom.
(d) Clothes spin round in the drum of a washing machine. (Sang W18)
7 Motion in a Circle
9
Q11. Assuming that the Earth goes around the Sun in a circle of radius 1.50 × 1011 m, find the centripetal
acceleration of the Earth. (Hutchings pg. 111)
Q12. In a model of the hydrogen atom, the electron moves round the nucleus in a circular orbit of radius
5.3 × 10-11 m. The speed of the electron is 2.2 × 106 m s-1. Calculate the centripetal acceleration of the
electron. (Mee pg. 255)
Q13. A satellite orbits the Earth 200 km above its surface. The acceleration towards the centre of the
Earth is 9.2 m s-2. The radius of the Earth is 6400 km. Calculate:
(a) the speed of the satellite, and
(b) the time to complete one orbit. (Mee pg. 255)
Q14. A satellite of mass 800 kg is orbiting the Moon in a circular path of radius 1760 km. The weight of
the satellite is 1300 N. Find the speed of the satellite and its period of rotation. (Hutchings pg. 114)
Q15. A toy train moves around a circular track of diameter 0.70 m, completing one revolution in
10 seconds. Calculate, for this train:
(a) the linear speed,
(b) the angular speed,
(c) the centripetal acceleration. (Mee pg. 186)
Q16. An electric motor has a rotor that rotates with an angular velocity of 50 revolutions per second.
The rotor has a radius of 12.0 cm.
(a) What is the centripetal acceleration of the rim of the rotor?
(b) How does the centripetal acceleration of the rotor vary through the body of the rotor from its centre
to its rim? (Hutchings pg. 112)
Q17. The maximum speed of the blades on rotary lawn-mowers is restricted to avoid danger from flying
stones. If the rate of rotation of the blade in one particular model is 3500 revolutions per minute and
the blade has a radius of 0.23 m, find
(a) The angular velocity of the blade;
(b) The linear velocity of the tip of the blade;
(c) The centripetal acceleration of the tip of the blade. (Hutchings pg. 112)
Q18. A centrifuge is required to give an acceleration of 1000g to a particle at a distance of 8.5 cm from
the axis of rotation. Find the necessary angular velocity of the centrifuge. (Hutchings pg. 113)
Q19. An object of mass of 2.0 kg rotates at constant speed in a horizontal circle of radius 5.0 m. The time
for one complete revolution is 3.0 s. What is the magnitude of the resultant force acting on the object?
Q20. An aeroplane is circling in the sky at a speed of 150 m s−1. The aeroplane describes a circle of
radius 20 km. For a passenger of mass 80 kg inside this aeroplane, calculate:
(a) her angular velocity,
(b) her centripetal acceleration,
(c) the centripetal force acting on her. (Sang W18)
7 Motion in a Circle
10
Q21. A car of mass 1000 kg is traveling around a bend on a level road. It completes the 120° in 15 s. The
radius of curvature of the bend is 250 m. Calculate the frictional force between the tyres and the road
surface.
Q22. An aircraft is flying along a straight horizontal path at 200 m s-1. The pilot tilts the wings through
an angle of 10°. What is the radius of circular path, which the plane will follow?
Q23. A pilot ‘banks’ the wings of his aircraft so as to travel at a speed of 390 km h-1 in a horizontal
circular path of radius 4.8 km. At what angle should he bank his aircraft in order to do this?
Q24. The diagram shows a stone tied to the end of a length of string. It is whirled round in a horizontal
circle of radius 80 cm.
The stone has a mass of 90 g and it completes 10 revolutions
in a time of 8.2 s.
(a) Calculate:
(i) the time taken for one revolution,
(ii) the distance travelled by the stone during one
revolution (this distance is equal to the circumference of the circle),
(iii) the speed of the stone as it travels in the circle,
(iv) the centripetal acceleration of the stone,
(v) the centripetal force on the stone.
(b) What provides the centripetal force on the stone?
(c) What is the angle between the acceleration of the stone and its velocity? (Sang W18)
Q25. A lump of clay of mass 300 g is placed close to the edge
of a spinning turntable. The centre of mass of the lump of clay
travels in a circle of radius 12 cm.
(a) The lump of clay takes 1.6 s to complete one revolution.
(i) Calculate the linear speed of the clay.
(ii) Calculate the frictional force between the clay and
the turntable.
(b) The maximum magnitude of the frictional force F between
the clay and the turntable is 70% of the weight of the clay. The speed of rotation of clay is slowly
increased.
Determine the speed of the clay when it just starts to slip off the turntable. (Sang W18)
Q26. A racing car of total mass 600 kg travels in a circle of radius 80 m at 30 m s-1 round a corner.
(a) Draw a diagram showing the forces acting on the car if:
(i) the track surface is horizontal and,
(ii) the track is banked so that there is no tendency for the car to skid sideways.
(b) Calculate the required frictional force in (a)(i) and the required angle of banking in (a)(ii).
(Hutchings pg. 114)
7 Motion in a Circle
11
Q27. A car of mass 820 kg travels at a constant speed of
32 m s−1 along a banked track. The track is banked at an
angle of 20° to the horizontal.
(a) The net vertical force on the car is zero. Use this to
show that the contact force R on the car is 8.56 kN.
(b) Use the answer from (a) to calculate the radius of the
circle described by the car. (Sang W18)
n
l
H
i
i
Q28. The maximum safe speed of a car rounding an unbanked corner is 20 m s-1 when the road is dry.
The maximum frictional force between the road surface and the wheels of the car is halved when the
road is wet. What is the maximum safe speed for the car to round the corner when the road is wet?
Q29. A grinding wheel of diameter 0.12 m spins horizontally
about a vertical axis, as shown in the diagram.
P is a typical grinding particle bonded to the edge of the wheel.
(a) If the rate of rotation is 1200 revolutions per minute,
calculate
(i) the angular velocity,
(ii) the acceleration of P,
(iii) the magnitude of the force acting on P if its mass is
1.0 × 10-4 kg.
(b) The maximum radial force at which P remains bonded to the wheel is 2.5 N.
Calculate the angular velocity at which P will leave the wheel if the rate of rotation is increased.
(Hutchings pg. 118)
Q30. A ball of mass 0.10 kg is attached to a string and swung in a vertical
circle of radius 0.50 m, as shown. Its speed at the top of the circle is 6.0 m s-1.
What is the tension in the string at this moment?
Q31. A stone of mass 120 g is fixed to one end of a light rigid rod.
The stone is whirled at a constant speed of 4.0 m s−1 in a vertical
circle of radius 80 cm.
Calculate the ratio:
ñóòôöõò öò úùó ûõü †ú °
.
ñóòôöõò öò úùó ûõü †ú ¢
(Sang W18)
7 Motion in a Circle
12
Q32. The figure below is a photograph of an amusement park ride in which a person of mass 63 kg
rotates in a vertical circle of radius 6.6 m. The time taken for one revolution is 3.2 s.
Find the force that is exerted by the structure on the rider when the rider is
(a) at the bottom of the circle and
(b) at the top of the circle. (Hutchings pg. 115)
Q33. The diagram shows a skateboarder of mass 70 kg who drops through a vertical height of 5.2 m.
The dip has a radius of curvature of 16 m.
(a) Assuming no energy losses due to air resistance or friction, calculate the speed of the skateboarder
at the bottom of the dip at point B. You may assume that the speed of the skateboarder at point A is
zero.
(b)
(i) Calculate the centripetal acceleration of the skateboarder at point B.
(ii) Calculate the contact force R acting on the skateboarder at point B. (Sang W18)