Homework 7
Due: December 4, 2014
Make any necessary assumptions for the following questions. If you do any
assumption explain its applicability.
1. Consider a point mass mattached to the end of a massless string. The
other end of the massless rod is fixed to a wall as shown in the figure.
Let θbe the angle that the rod makes with the vertical. Assume that
the mass is pulled to one side such that θ = π2 . Then it is released with
zero initial speed.
A
θ
m
(a) Using conservation of mechanical energy, calculate the kinetic energy and the speed of the mass m when it passes through the
bottom point (θ = 0)
(b) What is the moment of inertia of the mass m relative to rotations
around the point A?
(c) What is the angular velocity of the mass m as it goes through the
bottom point?
(d) What is the rotational energy of the mass m for rotations around
the point A as it goes through the bottom point?
(e) Compare the rotational energy that you calculated in the previous
part with the kinetic energy that you calculated in part (a).
2. Consider the example of a ring mass willing down the incline
(a) If the ring rolls a distance L before reaching level surface, how
long does it take to reach the level surface?
1
(b) Using the time dependence of angular velocity, what is its angular
velocity when it reaches the level surface? (in this part, use your
result from part (a))
(c) When the ring mass it rolling on the horizontal surface, what is
its rotational kinetic energy for rotations around an axis passing
through its centre of mass (CM)?
(d) When the ring mass is rolling on the horizontal surface, what is
the translation kinetic energy of the ring?
(e) When the ring is rolling without slipping, we also said that it can
be treated as if it is rotating around an axis that goes through the
contact point. This axis does not do any translational motions.
What is the rotational kinetic energy around this axis? Compare
it with the result that you would obtain from the previous two
parts.
(f) As the ring mass is rolling down, we had calculated how much
friction is acting on our system and shown that its magnitude is
1
Ff r = M g sin θ
2
(1)
Now assume that the a point mass M slides down the same incline.
The friction force acting on the mass is given by Eq. 1. (Note that
it is NOT equal to µN ). After it slides down by Lalong the incline,
what is the work done by friction?
(g) Compare the work done by friction in the previous part with the
rotational kinetic energy of rotation of the wheel about an axis
passing through its centre of mass.
3. Consider a 18 speed bicycle, 3 gears are rotating with the pedal and
6 gears are rotating with rear wheel. Consider that the ones rotating
with pedal, have radii 10 cm, 7 cm and 5 cm and the other ones have
radii 3 cm, 4 cm, 5 cm, 6 cm, 7 cm and 8 cm. Calculate the torque
on rear wheel gear for each different case if you apply to the pedal a
force of magnitude 343 N. The force is vertical to the pedal, and it has
a length of 17 cm. Explain why a bike needs that much speed, i.e. 18
speed.
2