Circular motion
1. In the following examples name the force that is providing the centripetal force and draw it
on the diagram.
(a) A runner running round a circular track.
(b) A car on a rollercoaster.
2. A 2kg mass travels in a circle of radius 50cm. If the time for one revolution is 2s calculate:
(a) The angular velocity of the mass
(b) The centripetal acceleration of the mass
(c) The centripetal force of the mass
3. A ball rolls around the inside of a vertical cylinder as shown. Indentify the force that stops it
from falling down.
Formulae
ω=2π/T
F=mv2/r=mω2r
© Chris Hamper, InThinking
www.physics-inthinking.co.uk
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Gravitational field
1. State Newton’s universal law of gravity.
2. Two masses are positioned as shown in the diagram.
Calculate the Force on the red one.
3. Define gravitational field strength.
4. Given that the mass of the moon is about 1/80 of the earth and its radius is ¼ estimate the
acceleration due to gravity on the surface of the moon.
Formulae
F=GMm/r2
G=6.7x10-11 m3kg-1s-2
© Chris Hamper, InThinking
www.physics-inthinking.co.uk
1
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QUESTIONS
Questions
1
A particle P is moving in a circle with uniform
speed. Draw a diagram to show the direction of
the acceleration a and velocity v of the particle
at one instant of time.
5
The Singapore Flyer is a large Ferris wheel
of radius 85 m that rotates once every
30 minutes.
P
2
State what provides the centripetal force that
causes a car to go round a bend.
3
State the centripetal force that acts on a particle
of mass m when it is travelling with linear
speed v along the arc of a circle of radius r.
4
(IB) At time t = 0 a car moves off from rest in a
straight line. Oil drips from the engine of the car
with one drop every 0.80 s. The position of the
oil drops on the road are drawn to scale on the
grid below such that 1.0 cm represents 4.0 m.
The grid starts at time t = 0.
a) Calculate the linear speed of a point on the
rim of the wheel of the Flyer.
b) (i) Determine the fractional change in the
weight of a passenger on the Flyer at
the top of the ride.
(ii) Explain whether the passenger has a
larger or smaller apparent weight at the
top of the ride.
c) The capsules need to rotate to keep the
floor of the cabin in the correct place.
Calculate the angular speed of the capsule
about its central axis.
Direction of motion
1.0 cm
6
a) (i) State the feature of the diagram that
indicates that the car accelerates at the
start of the motion.
a) Quito in Ecuador (14 minutes of arc south
of the Equator)
(ii) Determine the distance moved by the
car during the first 5.6 s of its motion.
b) The car then turns a corner at constant
speed. Passengers in the car who were
sitting upright feel as if their upper bodies
are being “thrown outwards”.
The radius of the Earth is 6400 km. Determine
the linear speed of a point on the ground at the
following places on Earth:
b) Geneva in Switzerland (46° north of the
Equator)
c) the South Pole.
7
(i) Identify the force acting on the car, and
its line of action, that enables the car to
turn the corner.
A school bus of total mass 6500 kg is carrying
some children to school.
a) During the journey the bus needs to travel
round in a horizontal curve of radius 150 m.
The dynamic coefficient of friction between
the tyres and the road surface is 0.7.
Estimate the maximum speed at which the
driver should attempt the turn.
(ii) Explain why the passengers feel as if
they are being thrown outwards.
265
3
6
CIR C UL A R MOT ION A N D GRAV ITATION
b) Later in the journey the driver needs to
drive across a curved bridge with a radius of
curvature of 75 m. Estimate the maximum
speed if the bus is to remain in contact with
the road.
8
A velodrome used for bicycle races has a
banking angle that varies continuously from 0°
to 60°. Explain how the racing cyclists use this
variation in angle to their advantage in a race.
11 Determine the distance from the centre of the
Earth to the point at which the gravitational field
strength of the Earth equals that of the Moon.
12 The ocean tides on the Earth are caused by the
tidal attraction of the Moon and the Sun on the
water in the oceans.
a) Calculate the force that acts on 1 kg of
water at the surface of the sea due its
attraction by the
Data needed for these questions:
(i) Moon
Radius of Earth = 6.4 Mm;
Mass of Earth = 6.0 × 1024 kg;
Mass of Moon = 7.3 × 1022 kg;
Mass of Sun = 2.0 × 1030 kg;
Earth–Moon distance = 3.8 × 108 m;
Sun–Earth distance = 1.5 × 1011 m;
G = 6.67 × 10–11 N m2 kg–2
(ii) Sun.
9
b) Optional – difficult. Explain why there are
two tides every day at many coastal points
on the Earth.
[Hint: there are two parts to the answer,
why a tide at all, and why two every day.]
Deduce how the radius R of the circular orbit of
a planet around a star of mass ms relates to the
period T of the orbit.
10 A satellite orbits the Earth at constant speed as
shown below.
satellite
13 There are two types of communication
satellite. One type of communication satellite
orbits over the poles at a distance from
the centre of the Earth of 7400 km; the
other type is geostationary with an orbital
radius of 36 000 km. Geostationary satellites
stay above one point on the equator whereas
polar-orbit satellites have an orbital time of
100 minutes.
Calculate:
a) the gravitational field strength at the
position of the polar-orbit satellite
b) the angular speed of a satellite in
geostationary orbit
Earth
c) the centripetal force acting on a
geostationary satellite of mass 1.8 × 103 kg.
a) Explain why, although the speed of
the satellite is constant, the satellite is
accelerating.
b) Discuss whether or not the gravitational
force does work on the satellite.
266
4
Topic 6.1a Circular Motion Problems
Conceptual Questions
(These questions are not in an IB style but instead designed to check your understanding of the concept of this topic. You should
try your best to appropriately communicate your answer using prose)
1. Sometimes people say that water is removed from clothes in a spin-dryer by centrifugal force
throwing the water outward. What is wrong with this statement?
2. A girl is whirling a ball on a string around her head in a horizontal plane. She wants to let go at
precisely the right time so that the ball will hit a target on the other side of the yard. When should
she let go of the string?
3. A bucket of water can be whirled in a vertical circle without the water spilling out, even at the top
of the circle when the bucket is upside-down. Explain.
5
Topic 6.1a Circular Motion Problems
Calculation-based Questions
1. Calculate the centripetal acceleration of the Earth in its orbit around the Sun, and the net force
exerted on the Earth. What exerts this force on the Earth? Assume that the Earth’s orbit is a circle
of radius 1.5x1011m. You may need to look other constants up on the Internet or data booklet.
[3 marks]
2. A horizontal force of 210N is exerted on a 2.0kg discus as it rotates uniformly in a horizontal circle
(at arm’s length) of radius 0.90m. Calculate the speed of the discus.
[2 marks]
3. Suppose the space shuttle in orbit 400km from the Earth’s surface, and circles the Earth about once
every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your
answer in terms of g, the gravitational acceleration at the Earth’s surface. You may need to look up
some other constants on the Internet.
[3 marks]
6
4. A flat puck (mass M) is rotated in a circle on a frictionless air-hockey tabletop, and is held in its orbit
by a light cord connected to a dangling block (mass m) through a central hole as shown below.
Show that the speed of the puck is given by
𝑣=#
𝑚𝑔𝑅
𝑀
[2 marks]
5. A 0.45kg ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3m on a
frictionless horizontal surface. If the cord will break when the tension in it exceeds 75N, what is the
maximum speed the ball can have?
[2 marks]
7
Topic 6.2 Gravitation Problems
Conceptual Questions
(These questions are not in an IB style but instead designed to check your understanding of the concept of this topic. You should
try your best to appropriately communicate your answer using prose)
1. Does an apple exert a gravitational force on the Earth? If so, how large a force? Consider an apple
attached to the tree and also falling.
2. Will an object weigh more at the equator or at the poles? Explain.
Calculation Based
3. Calculate the force of Earth’s gravity on a spacecraft 12,800km (2 Earth radii) above the Earth’s
surface if its mass is 1350kg. [2 marks]
4. At the surface of a certain planet, the gravitational acceleration g has a magnitude of 12m/s2. A
21.0kg brass ball is transported to this planet. What is:
a. The mass of the brass ball on the Earth and on the planet. [1 mark]
b. The weigh of the brass ball on the Earth and on the planet. [1 mark]
8
5. A hypothetical planet has a radius 1.5 times that of the Earth, but has the same mass. What is the
acceleration due to gravity near the surface? (Hint: use the Internet or your data book to find the constants).
[2 marks]
6. In the diagram shown, two point particles are fixed on an x-axis separated by a distance d. Particle
A has mass mA and particle B has mass 3.00mA. A third particle C, of mass 75.0mA is to be placed on
the axis and near particles A and B. In terms of distance d, at what x-coordinate should C be placed
so that the net gravitational force on particle A from particles B and C is zero?
7. (a) What will an object weigh on the Moon’s surface if it weights 100N on the Earth’s surface? (b)
How many Earth radii must this same object be from the centre of the Earth if it is to weigh the
same as it does on the Moon?
9
8. Two concentric spherical shells with uniformly distributed masses M1 and M2 are situated as
shown in the diagram. Find the magnitude of the net gravitational force on a particle of mass m,
due to the shells, when the particle is located at radial distance (a) a, (b) b and (c) c.
10
Exam-style questions
1 A child is sitting at the edge of a merry-go-round. The arrow shows the velocity of the child. At the instant shown,
he releases a ball onto the ground.
child
Which is the path of the ball according to a stationary observer on the ground?
A
B
C
D
2 In which of the following examples of circular motion is the centripetal acceleration experienced by the particle
the largest? In each case the arrows represent speed.
A
B
C
D
3 A horizontal disc rotates about a vertical axis through the centre of the disc.
Two particles X and Y are placed on the disc.
The particles do not move relative to the disc. Which is correct about the
angular speed and the linear speed v of X and Y?
X
Y
v
A
B
C
D
same
same
different
different
same
different
same
different
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6 CIRCULAR MOTION AND GRAVITATION
265
4 In the diagram for question 3 the ratio of distances of Y to X is 2. What is the ratio of the acceleration of Y to
that of X?
A
1
4
B
1
2
C 2
D 4
5 A particle of mass m moves with speed v along a hill that may be assumed to be part of a circle of radius r.
v
What is the reaction force on the particle at the highest point on the hill?
A mg
B mg +
mv 2
r
C mg −
mv 2
r
D
mv 2
− mg
r
6 A particle moves with speed v in a circular orbit of radius r around a planet. The particle is now moved to another
circular orbit of radius 2r. The new orbital speed is:
A
v
2
B
v
√2
C v √2
D 2v
7 The mass of a landing module on the Moon is 2000 kg. The gravitational field strength on the Moon is one-sixth
that on Earth. What is the weight of the landing module on Earth?
A 330 N
B 2000 N
C 12 000 N
D 20 000 N
8 A planet has double the mass of Earth and half its radius. What is the gravitational field strength on the surface of
this planet?
A 10 N kg−1
B 20 N kg−1
C 40 N kg−1
D 80 N kg−1
9 A satellite orbits the Earth in a circular orbit. The only force on the satellite is the gravitational force from the
Earth. Which of the following is correct about the acceleration of the satellite?
A
B
C
D
It is zero.
It is constant in magnitude and direction.
It is constant in magnitude but not in direction.
It is not constant in magnitude or direction.
10 The two spherical bodies in the diagram have the same radius but the left mass has twice the mass of the other.
At which point could the net gravitational field of the two masses have the greatest magnitude?
2M
A
266
M
B
C
D
12
11 A horizontal disc of radius 45 cm rotates about a vertical axis through its centre. The disc makes one full
revolution in 1.40 s. A particle of mass 0.054 kg is placed at a distance of 22 cm from the centre of the disc.
The particle does not move relative to the disc.
a On a copy of the diagram draw arrows to represent the velocity and acceleration of the particle.
b Calculate the angular speed and the linear speed of the particle.
c The coefficient of static friction between the disc and the particle is 0.82. Determine the largest distance
from the centre of the disc where the particle can be placed and still not move relative to the disc.
d The particle is to remain at its original distance of 22 cm from the centre of the disc.
i Determine the maximum angular speed of the disc so that the particle does not move relative to
the disc.
ii The disc now begins to rotate at an angular speed that is greater than the answer in d i. Describe
qualitatively what happens to the particle.
[2]
[2]
[3]
[2]
[2]
12 A block of mass of 5.0 kg is attached to a string of length 2.0 m which is initially horizontal. The mass is then
released and swings as a pendulum. The diagram shows the mass falling to the position where the string is in the
vertical position.
m
m
a
b
c
d
Calculate the speed of the block when the string is in the vertical position.
Deduce the acceleration of the block.
On a copy of the diagram, draw arrows to represent the forces on the block.
For when the string is in the vertical position:
i state and explain whether the block is in equilibrium
ii calculate the tension in the string.
13
[2]
[1]
[2]
[2]
[2]
6 CIRCULAR MOTION AND GRAVITATION
267
13 A particle of mass m is attached to a string of length L whose other end is attached to the ceiling, as shown in the
diagram. The particle moves in a horizontal circle making an angle of with the vertical. Air resistance may be
neglected.
θ
m
a On a copy of the diagram draw arrows to represent the forces on the particle.
b State and explain whether the particle is in equilibrium.
c The linear speed of the particle is v and its angular speed is . Show that:
i v=
gL sin2
cos
[2]
[2]
[2]
g
L cos
d The length of the string is 45 cm and = 60°. Use the answer in c to evaluate:
i the linear speed
ii the angular speed of the particle.
Air resistance may no longer be neglected.
e Suggest the effect of air resistance on:
i the linear speed of the particle
ii the angle the string makes with the vertical
iii the angular speed of the particle.
ii ω =
[2]
[1]
[1]
[1]
[1]
[1]
14 A marble rolls from the top of a big sphere, as shown in the diagram.
θ
a Show that when the marble has moved so that the line joining it to the centre of the sphere is , its speed
[3]
is given by v = √2gR(1 − cos ). (Assume a very small speed at the top.)
b Deduce that at that instant, the normal reaction force on the marble from the sphere is given by
[3]
N = mg(3 cos − 2).
[1]
c Hence determine the angle at which the marble loses contact with the sphere.
268
14
15 Consider two spherical bodies of mass 16M and M as in the diagram.
d
M
16M
There is a point P somewhere on the line joining the masses where the gravitational field strength is zero.
a Determine the distance of point P from the centre of the bigger mass in terms of d, the centre-to-centre
distance separating the two bodies.
b Draw a graph to show the variation of the gravitational field strength g due to the two masses with the
distance x from the centre of the larger mass.
c A small point mass m is placed at P.
i State the force on m.
ii The small mass m is slightly displaced to the left of P. State and explain whether the net force on the
point mass will be directed to the left or to the right.
d Describe qualitatively the motion of the point mass after it has been displaced to the left of P.
[3]
[2]
[1]
[2]
[2]
16 A satellite is in a circular orbit around a planet of mass M, as shown in the diagram.
i On a copy of the diagram draw arrows to represent the velocity and acceleration of the satellite.
ii Explain why the satellite has acceleration even though its speed is constant.
b Show that the angular speed is related to the orbit radius r by r3 2 = GM.
c Because of friction with the upper atmosphere, the satellite slowly moves into another circular orbit with
a smaller radius than the answer in b. Suggest the effect of this on the satellite’s:
i angular speed
ii linear speed.
d Titan and Enceladus are two of Saturn’s moons. Data about these moons are given in the table.
a
Moon
Orbit radius / m
Titan
1.22 × 109
Enceladus
2.38 × 108
[2]
[2]
[3]
[1]
[1]
Angular speed / rad s−1
5.31 × 10−5
i Determine the mass of Saturn.
ii Determine the period of revolution of Titan in days.
15
[2]
[3]
6 CIRCULAR MOTION AND GRAVITATION
269
?
Test yourself
1 a Calculate the angular speed and linear speed of
a particle that completes a 3.50 m radius circle
in 1.24 s.
b Determine the frequency of the motion.
2 Calculate the centripetal acceleration of a body
that moves in a circle of radius 2.45 m making
3.5 revolutions per second.
3 The diagram shows a mass moving on a circular
path of radius 2.0 m at constant speed 4.0 m s−1.
B
A
a Calculate the magnitude and direction of the
average acceleration during a quarter of a
revolution (from A to B).
b Calculate the centripetal acceleration of the mass.
4 An astronaut rotates at the end of a test machine
whose arm has a length of 10.0 m, as shown in the
diagram. The acceleration she experiences must
not exceed 5g (take g = 10 m s−2 ). Determine the
maximum number of revolutions per minute of
the arm.
6 Estimate the length of the day if the centripetal
acceleration at the equator due to the spinning
Earth was equal to the acceleration of free fall
(g = 9.8 m s−2 ).
7 A neutron star has a radius of 50.0 km and
completes one revolution every 25 ms.
a Calculate the centripetal acceleration
experienced at the equator of the star.
b The acceleration of free fall at the surface of
the star is 8.0 × 1010 m s−2. State and explain
whether a probe that landed on the star could
stay on the surface or whether it would be
thrown off .
8 The Earth (mass = 6.0 × 1024 kg) rotates around
the Sun in an orbit that is approximately circular,
with a radius of 1.5 × 1011 m.
a Estimate the orbital speed of the Earth around
the Sun.
b Determine the centripetal acceleration
experienced by the Earth.
c Deduce the magnitude of the gravitational
force exerted on the Sun by the Earth.
9 A plane travelling at a speed 180 m s−1 along a
horizontal circle makes an angle of = 35° to
the horizontal. The lift force L is acting in the
direction shown. Calculate the radius of the circle.
L
10 m
θ
5 A body of mass 1.00 kg is tied to a string and
rotates on a horizontal, frictionless table.
a The length of the string is 40.0 cm and the
speed of revolution is 2.0 m s−1. Calculate the
tension in the string.
b The string breaks when the tension exceeds
20.0 N. Determine the largest speed the mass
can rotate at.
c The breaking tension of the string is 20.0 N
but you want the mass to rotate at 4.00 m s−1.
Determine the shortest length string that can
be used.
10 A cylinder of radius 5.0 m rotates about its
vertical axis. A girl stands inside the cylinder with
her back touching the side of the cylinder. The
floor is suddenly lowered but the girl stays ‘glued’
to the wall. The coefficient of friction between
the girl and the wall is 0.60.
a Draw a free body diagram of the forces on
the girl.
b Determine the minimum number of
revolutions per minute for which the girl does
not slip down the wall.
16
6 CIRCULAR MOTION AND GRAVITATION
257
11 A loop-the-loop machine has radius r of 18 m.
13 The ball shown in the diagram is attached to
a rotating pole with two strings. The ball has a
mass of 0.250 kg and rotates in a horizontal circle
at a speed of 8.0 m s−1. Determine the tension in
each string.
r
v=?
a Calculate the minimum speed with which a
cart must enter the loop so that it does not fall
off at the highest point.
b Predict the speed at the top in this case.
12 The diagram shows a horizontal disc with a hole
through its centre. A string passes through the
hole and connects a mass m on top of the disc
to a bigger mass M that hangs below the disc.
Initially the smaller mass is rotating on the disc
in a circle of radius r. Determine the speed of m
be such that the big mass stands still.
m
1.0 m
0.50 m
0.50 m
1.0 m
14 In an amusement park ride a cart of mass 300 kg
and carrying four passengers each of mass 60 kg
is dropped from a vertical height of 120 m along
a frictionless path that leads into a loop-the-loop
machine of radius 30 m. The cart then enters
a straight stretch from A to C where friction
brings it to rest after a distance of 40 m.
A
B
h
C
R
M
a Determine the velocity of the cart at A.
b Calculate the reaction force from the seat of
the cart onto a passenger at B.
c Determine the acceleration experienced by
the cart from A to C (assumed constant).
258
17
?
Test yourself
15 Calculate the gravitational force between:
a the Earth and the Moon
b the Sun and Jupiter
c a proton and an electron separated by 10−10 m.
16 A mass m is placed at the centre of a thin, hollow,
spherical shell of mass M and radius r, shown in
diagram a.
22 The diagram shows point P is halfway between
the centres of two equal spherical masses that are
separated by a distance of 2 × 109 m. Calculate
the gravitational field strength at point P and
state the direction of the gravitational field
strength at point Q.
2 × 109 m
M
M
r
P
r
m
m
2r
3 × 1022 kg
a
17
18
19
20
21
264
109 m
3 × 1022 kg
b
a Determine the gravitational force the mass m
experiences.
b Determine the gravitational force m exerts
on M.
A second mass m is now placed a distance of 2r
from the centre of the shell, as shown in
diagram b.
c Determine the gravitational force the mass
inside the shell experiences.
d Suggest what gravitational force is
experienced by the mass outside the shell.
Stars A and B have the same mass and the radius
of star A is nine times larger than the radius of
star B. Calculate the ratio of the gravitational
field strength on star A to that on star B.
Planet A has a mass that is twice as large as the
mass of planet B and a radius that is twice as
large as the radius of planet B. Calculate the ratio
of the gravitational field strength on planet A to
that on planet B.
Stars A and B have the same density and star A
is 27 times more massive than star B. Calculate
the ratio of the gravitational field strength on star
A to that on star B.
A star explodes and loses half its mass. Its radius
becomes half as large. Determine the new
gravitational field strength on the surface of the
star in terms of the original one.
The mass of the Moon is about 81 times less
than that of the Earth. Estimate the fraction of
the distance from the Earth to the Moon where
the gravitational field strength is zero. (Take into
account the Earth and the Moon only.)
Q
23 A satellite orbits the Earth above the equator
with a period equal to 24 hours.
a Determine the height of the satellite above
the Earth’s surface.
b Suggest an advantage of such a satellite.
24 The Hubble Space Telescope is in orbit around
the Earth at a height of 560 km above the Earth’s
surface. Take the radius and mass of the Earth to
be 6.4 × 106 m and 6.0 × 1024 kg, respectively.
a Calculate Hubble’s speed.
b In a servicing mission, a Space Shuttle spotted
the Hubble telescope a distance of 10 km
ahead. Estimate how long it took the Shuttle
to catch up with Hubble, assuming that the
Shuttle was moving in a circular orbit just
500 m below Hubble’s orbit.
25 Assume that the force of gravity between two
Gm1m2
where n is
point masses is given by F =
rn
a constant.
a Derive the law relating period to orbit radius
for this force.
b Deduce the value of n if this law is to be
identical with Kepler’s third law.
18
Topic 6 (New) [54 marks]
An electron moves in circular motion in a uniform magnetic field.
The velocity of the electron at point P is 6.8 × 10 5 m s –1 in the direction shown.
The magnitude of the magnetic field is 8.5 T.
1a. State the direction of the magnetic field.
[1 mark]
1b. Calculate, in N, the magnitude of the magnetic force acting on the electron.
[1 mark]
1c. Explain why the electron moves at constant speed.
[1 mark]
Explain why the electron moves on a circular19
path.