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Test yourself
1 A disc has an initial angular velocity 3.5 rad s−1 6 A uniform plank of length 2.0 m and weight 58 N
and after 5.0 s the angular velocity increases to is supported horizontally by a cable attached to a
15 rad s−1. Determine the angle through which vertical wall.
the disc has turned during that 5.0 s.
2 A body rotates about an axis with an angular
velocity of 5.0 rad s−1. The angular acceleration is
2.5 rad s−2. Calculate the body’s angular velocity
after it has turned through an angle of 54 rad.
3 A body rotates with an initial angular velocity
of 3.2 rad s−1. The angular velocity increases to
12.4 rad s−1 in the course of 20 full revolutions.
Calculate the angular acceleration.
3.0 m
4 A uniform plank of weight 450 N and length
5.0 m is supported at both ends. A block of
weight 120 N is placed at a distance of 2.0 m from
the left end. Calculate the force at each support.
120 N
2.0 m
2.0 m
5.0 m
5 A uniform plank of mass 30 kg and length 4.0 m Calculate a the tension in the cable, and b the
is supported at its left end and at a point 0.80 m magnitude and direction of the force exerted by
from the middle. the wall on the rod.
7 A cylinder of mass 5.0 kg and radius 0.20 m is
attached to an axle parallel to its axis and through
its centre of mass. A constant force of 6.5 N acts
on the cylinder as shown in below. Find the
2.0 m 0.80 m x angular speed of the cylinder after 5.0 s.
Calculate the largest distance x to which a boy
of mass 40 kg can walk without tipping the rod
over.
F = 6.5 N
16 B ENGINEERING PHYSICS PHYSICS FOR THE IB DIPLOMA © CAMBRIDGE UNIVERSITY PRESS 2015
8 A point mass, a sphere, a cylinder and a ring 11 A rod of length L = 1.20 m and mass M = 3.00 kg
each have mass M. The solid bodies have the is free to move about a fi xed axis at its left end.
same radius R. The four bodies are released Its moment of inertia about this axis is given
from the same position on an inclined plane by 13ML 2.
(with the ring upright, so it rolls). Determine
the order, from least to greatest, of their speeds
A
when they reach level ground. Assume rolling
without slipping.
9 A disc of mass 12 kg and radius 0.35 m is
spinning with angular velocity 45 rad s−1.
a Determine how many revolutions per minute
(rpm) the disc is making.
b A force is applied to the rim of the disc, B
tangential to the disc’s circumference.
Determine the magnitude of this force if the
disc is to stop spinning after 4.0 s. The rod is is released from rest in the horizontal
c Calculate the number of revolutions it makes position A.
before stopping. a Calculate the initial angular acceleration of
10 A uniform sphere rotates about a fi xed the rod.
horizontal axis through the edge of the sphere, b State and explain whether the angular
as shown below. The axis is at the height of the acceleration is constant in magnitude as the
initial position of the sphere’s centre of mass. rod rotates.
The moment of inertia of the sphere about this c Calculate the angular velocity of the rod as it
axis is I = 75MR 2. The sphere is released from rest moves past the vertical position B.
at position A. 12 A horizontal force F is applied to the surface of
a cylinder of mass M and radius R. The force is
applied a vertical distance d above its centre of
mass, as shown below.
A
R
d
F
B Determine d as a fraction of R such that the
cylinder rolls without slipping. The moment of
a Find an expression for the initial angular inertia of a cylinder about its central axis is
1 2
acceleration of the sphere. 2MR .
b State and explain whether the angular
acceleration is constant in magnitude as the
sphere rotates.
c Find an expression for the angular velocity of
the sphere as it moves past position B.
PHYSICS FOR THE IB DIPLOMA © CAMBRIDGE UNIVERSITY PRESS 2015 B ENGINEERING PHYSICS 17
13 A block of mass m hangs from the end of a 16 A star of mass M and radius R explodes radially
string that goes over a pulley of radius R, and is and symmetrically. The star is left with a mass
1 1
connected to another block of mass M that rests of 10 M and a radius of 50 R. Calculate the ratio
on a horizontal table. of the star’s final angular velocity to its initial
angular velocity.
M 17 A battery-driven toy car of mass 0.18 kg is
placed on a circular track that is part of a
horizontal ring with radius 0.50 m and moment
of inertia 0.20 kg m 2 relative to its vertical axis.
The ring can rotate about this axis without
friction.
a Assuming that the pulley is massless and that
there are no frictional forces, calculate the
acceleration of each block when the smaller
block is released.
b The pulley is now assumed to have mass
M. The small block is again released and
the pulley turns as the blocks move. Again
calculate the acceleration of each block. (Hint:
the tensions in the two strings are different.)
14 A horizontal disc, with radius 0.80 m and
moment of inertia 280 kg m 2 about its vertical
axis, rotates about this axis at 320 revolutions
The car is started and its speed is measured to
per minute. The disc is brought to rest in 12 s.
be 0.80 m s−1 relative to the ground. Calculate
Calculate a the work done, and b the power
the angular speed of the ring.
developed in stopping the disc.
15 Two identical rods, X and Y, each of length
1.20 m and mass 2.40 kg, are made to rotate
about different vertical axes, as shown below
each with angular velocity 4.50 rad s−1.
X
1
I= ML2
12
Y
1
I = ML2
3
Calculate a the kinetic energy and b the
angular momentum of X and of Y.
18 B ENGINEERING PHYSICS PHYSICS FOR THE IB DIPLOMA © CAMBRIDGE UNIVERSITY PRESS 2015
