?
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 θ = 35° to
the horizontal. The lift force L is acting in the
direction shown. Calculate the radius of the circle.
L
10 m
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.
θ
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.
6 CIRCULAR MOTION AND GRAVITATION
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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).
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