112
3 Rotational Kinematics
3.2.2 Motion in a Vertical Plane
3.26 A particle is placed at the highest point of a smooth sphere of radius R and is
given an infinitesimal displacement. At what point will it leave the sphere?
[University of Cambridge]
3.27 A small sphere is attached to a fixed point by a string of length 30 cm and
whirls round in a vertical circle under the action of gravity at such a speed that
the tension in the string when the sphere is at the lowest point is three times
the tension when the sphere is at its highest point. Find the speed of the sphere
at the highest point.
[University of Cambridge]
3.28 A light rigid rod of length L has a mass m attached to its end, forming a simple
pendulum. The pendulum is put in the horizontal position and released from
rest. Show that the tension in the suspension will be equal to the magnitude of
weight at an angle θ = cos −1 (1/3) with the vertical.
3.29 In a hollow sphere of diameter 20 m in a circus, a motorcyclist rides with
sufficient speed in the vertical plane to prevent him from sliding down. If the
coefficient of friction is 0.8, find the minimum speed of the motorcyclist.
3.30 The bob of a pendulum of mass m and length L is displaced 90 ◦ from the
vertical and gently released. What should be the minimum strength of the
string in order that it may not break upon passing through the lowest point?
3.31 The bob of a simple pendulum of length L is deflected through a small arc s
from the equilibrium position and released. Show that when it passes through
the equilibrium position its velocity will be s
√
g/L, where g is the acceleration due to gravity.
3.32 A simple pendulum of length 1.0 metre with a bob of mass m swings with an
angular amplitude of 60 ◦ . What would be the tension in the string when its
angular displacement is 45 ◦ ?
3.33 The bob of a pendulum is displaced through an angle θ with the vertical line
and is gently released so that it begins to swing in a vertical circle. When
it passes through the lowest point, the string experiences a tension equal to
double the weight of the bob. Determine θ .
3.2.3 Loop-the-Loop
3.34 The bob of a simple pendulum of length 1.0 m has a velocity of 6 m/s when
it is at the lowest point. At what height above the centre of the vertical circle
will the bob leave the path?
3 Rotational Kinematics
3.2.2 Motion in a Vertical Plane
3.26 A particle is placed at the highest point of a smooth sphere of radius R and is
given an infinitesimal displacement. At what point will it leave the sphere?
[University of Cambridge]
3.27 A small sphere is attached to a fixed point by a string of length 30 cm and
whirls round in a vertical circle under the action of gravity at such a speed that
the tension in the string when the sphere is at the lowest point is three times
the tension when the sphere is at its highest point. Find the speed of the sphere
at the highest point.
[University of Cambridge]
3.28 A light rigid rod of length L has a mass m attached to its end, forming a simple
pendulum. The pendulum is put in the horizontal position and released from
rest. Show that the tension in the suspension will be equal to the magnitude of
weight at an angle θ = cos −1 (1/3) with the vertical.
3.29 In a hollow sphere of diameter 20 m in a circus, a motorcyclist rides with
sufficient speed in the vertical plane to prevent him from sliding down. If the
coefficient of friction is 0.8, find the minimum speed of the motorcyclist.
3.30 The bob of a pendulum of mass m and length L is displaced 90 ◦ from the
vertical and gently released. What should be the minimum strength of the
string in order that it may not break upon passing through the lowest point?
3.31 The bob of a simple pendulum of length L is deflected through a small arc s
from the equilibrium position and released. Show that when it passes through
the equilibrium position its velocity will be s
√
g/L, where g is the acceleration due to gravity.
3.32 A simple pendulum of length 1.0 metre with a bob of mass m swings with an
angular amplitude of 60 ◦ . What would be the tension in the string when its
angular displacement is 45 ◦ ?
3.33 The bob of a pendulum is displaced through an angle θ with the vertical line
and is gently released so that it begins to swing in a vertical circle. When
it passes through the lowest point, the string experiences a tension equal to
double the weight of the bob. Determine θ .
3.2.3 Loop-the-Loop
3.34 The bob of a simple pendulum of length 1.0 m has a velocity of 6 m/s when
it is at the lowest point. At what height above the centre of the vertical circle
will the bob leave the path?