114
3 Rotational Kinematics
3.40 A block is allowed to slide down a frictionless track freely under gravity. The
track ends in a circular loop of radius R. Show that the minimum height from
which the block must start is 2.5R so that it completes the circular track.
3.41 A nail is located at a certain distance vertically below the point of suspension
of a simple pendulum. The pendulum bob is released from a position where the
string makes an angle 60 ◦ with the vertical. Calculate the distance of the nail
from the point of suspension such that the bob will just perform revolutions
with the nail as centre. Assume the length of the pendulum to be 1 m.
[Indian Institute of Technology 1975]
3.42 A test tube of mass 10 g closed with a cork of mass 1 g contains some ether.
When the test tube is heated, the cork flies out under the pressure of the ether
gas. The test tube is suspended by a weightless rigid bar of length 5 cm. What
is the minimum velocity with which the cork would fly out of the test tube so
that the test tube describes a full vertical circle about the point of suspension?
Neglect the mass of ether.
[Indian Institute of Technology 1969]
3.43 A car travels at a constant speed of 14.0 m/s round a level circular bend of
radius 45 m. What is the minimum coefficient of static friction between the
tyres and the road in order for the car to go round the bend without skidding?
[University of Manchester 2008]
3.3 Solutions
3.3.1 Motion in a Horizontal Plane
3.1 x = a cos t
(1)
y = a sin t
(2)
z = t
(3)
Squaring (1) and (2) and adding
x
2
+ y
2
= a
2
(cos
2 t + sin
2 t) = a
2
which is the equation of a circle.
Since z = t, the circular path drifts along the z-axis so that the path is a helix.
3.2 a =
dv
dt
= k
2 rt
2
v =
dv = k
2 r
t
2 dt =
k 2 r t 3
3
Power, P = Fv = mav = mk
2 r t
2 k 2 r t 3
3
=
mk 4 r 2 t 5
3
3 Rotational Kinematics
3.40 A block is allowed to slide down a frictionless track freely under gravity. The
track ends in a circular loop of radius R. Show that the minimum height from
which the block must start is 2.5R so that it completes the circular track.
3.41 A nail is located at a certain distance vertically below the point of suspension
of a simple pendulum. The pendulum bob is released from a position where the
string makes an angle 60 ◦ with the vertical. Calculate the distance of the nail
from the point of suspension such that the bob will just perform revolutions
with the nail as centre. Assume the length of the pendulum to be 1 m.
[Indian Institute of Technology 1975]
3.42 A test tube of mass 10 g closed with a cork of mass 1 g contains some ether.
When the test tube is heated, the cork flies out under the pressure of the ether
gas. The test tube is suspended by a weightless rigid bar of length 5 cm. What
is the minimum velocity with which the cork would fly out of the test tube so
that the test tube describes a full vertical circle about the point of suspension?
Neglect the mass of ether.
[Indian Institute of Technology 1969]
3.43 A car travels at a constant speed of 14.0 m/s round a level circular bend of
radius 45 m. What is the minimum coefficient of static friction between the
tyres and the road in order for the car to go round the bend without skidding?
[University of Manchester 2008]
3.3 Solutions
3.3.1 Motion in a Horizontal Plane
3.1 x = a cos t
(1)
y = a sin t
(2)
z = t
(3)
Squaring (1) and (2) and adding
x
2
+ y
2
= a
2
(cos
2 t + sin
2 t) = a
2
which is the equation of a circle.
Since z = t, the circular path drifts along the z-axis so that the path is a helix.
3.2 a =
dv
dt
= k
2 rt
2
v =
dv = k
2 r
t
2 dt =
k 2 r t 3
3
Power, P = Fv = mav = mk
2 r t
2 k 2 r t 3
3
=
mk 4 r 2 t 5
3