Answers to survey
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- 1. Bouncing pingpong ball on top of golf ball
- The golf ball and pingpong ball fall together. When the golf
ball hits the ground at speed v, it rebounds and moves upwards at the
same speed, and collides with the pingpong ball coming down at speed
Transform to the reference frame of the golf ball, which sees a
pingpong ball approaching at 2v. The golf ball is infinitely massive
compared to the pingpong ball, so the pingpong ball bounces off and
moves upwards at speed 2v, relative to the golf ball.
Now transform back to the lab frame: The pingpong ball is moving
upwards at 3v. Since energy, and therefore height, is proportional to
v squared, the pingpong ball reaches to 9 times the initial height: to
- 2. Disc and hoop rolling down the plane
- If the objects were sliding without friction,
both would accelerate at the same rate,
with all the gravitational potential energy being converted into
kinetic energy of translation.
But these objects are rolling (the angular velocity of rolling is
increasing, in proportion to the linear velocity), so some of the
gravitational potential energy is converted into rotational kinetic
energy, leaving less available for linear acceleration.
The hoop's mass is more concentrated towards the edge than disc's is.
So compared to the disc, a larger fraction of the hoop's energy is
rotational: It's linear acceleration is less than the disc's. This
result does not depend on the masses or diameters of the objects. For
example, if you double the mass of the hoop, you double the
translational and rotational energies, but you do not change their
A way to see this difference is to think of extreme cases:
an extreme case of a hoop is a dumbbell with its mass
extending beyond the rolling radius.
- 3. Pendulum period versus amplitude
- The simple pendulum's equation of motion is
x'' = -sin(x).
For small amplitudes, the period is nearly independent of
amplitude, because the small-angle approximation replaces sin(theta)
with theta. Just having to make the approximation means that the
period cannot be independent of the amplitude, which rules out the
This approximation makes a *fractional* error of order theta squared.
The period should show the same effect. So the period versus
amplitude graph should be flat near theta=0 (just as a parabola is
flat near the origin), which rules out the two straight-line graphs
(one going up, one going down).
The choice remains between the two curved graphs, one with period
increasing with amplitude, the other with period decreasing. Think of
an extreme case: When the amplitude is 180 degrees, the period is
infinite (the pendulum stays exactly balanced forever). So the period
should increase with amplitude. Only one graph passes all these tests.
- 4. Merry-go-round
- In the earth's frame, once the ball is thrown there are no
sideways forces on it, so its path looks straight (as seen from
above). A straight path as seen from the earth looks curved
as seen from the merry-go-round.
- 5. Slinky
This question is most easily answered by dimensional analysis.
Assume the travel time tau is a function of the length l,
the spring constant k, and the mass m.
tau = f( m,l,k ).
f must have dimensions of time.
What dimensionless groups can we make from tau,m,l,k? Only
(tau/sqrt(m/k)). There is no group involving l.
So tau must be a function of the form
tau = sqrt(m/k) * constant.
There can be no l-dependence.
An alternative brute-force solution:
A slinky is a string,
with wave speed sqrt(F/lambda) where F is the tension and lambda is
the mass per unit length. If you double the length, you double the
F and halve lambda: The wave speed doubles. But so has the length;
the time for a wave to cross the slinky stays the same.
- 6. Sliding bead on a rotating wire
- Consider a limiting case: the wire rotating in a plane (theta=90).
Then, no matter where the bead starts, it'll fly off to infinity.
That behavior is also possible with a tilted wire, if the bead starts
far enough away: The "centrifugal force" will make it move still
farther, where it will move faster, where the greater centrifugal
force will drive it farther outwards, off to infinity.
Another behavior is also possible. If the bead starts too close to
the origin, then it moves too slowly for the centrifugal force to beat
gravity: The bead will fall inwards.
It then rotates more slowly,
where the centrifugal force will be even weaker, and gravity will keep
winning. The bead will head straight for the origin.
case that helps here is the case where the wire is vertical.)
There is a critical starting point from where the bead will stay
fixed, where gravity balances the centrifugal force. The arguments
above show why this equilibrium point is unstable.
- 7. Satellite in orbit
- In elliptical orbits, the energy and period are both
directly related to the length of the semi-major axis.
The angular momentum is related to the semi-latus rectum
(The semi-latus rectum is the line
drawn perpendicular to the major axis, through the force centre.)
The impulse increases the total energy of
the satellite, so the major axis (the long axis) increases. All three
choices had a longer major axis.
But the impulse also
increases the angular momentum, so the semi-latus rectum increases as
well. Only orbit C is consistent.
- 8. Wood block
- Doubling the thickness doubles the mass, which decreases
frequency; but doubling also increases stiffness,
which increases frequency. Which effect wins?
The musical note comes from the block bending slightly.
The lowest-frequency note corresponds to bending
(the same way a diving board bends). There are two nodes, located
roughly at 1/4 and 3/4 of the length of the block.
The mode looks like this, if the long axis is vertical:
| ( | ) | ( |
For a fixed
curvature of the block, let's compare the potential and kinetic energy
stored in a double-thickness block with that stored in the original
The kinetic energy is easy: The mass doubled so the kinetic energy
The potential energy is trickier. A material is a bunch of springs
(bonds), where potential energy is proportional to extension squared.
Doubling the thickness doubles the typical extension, so the energy
stored in each spring goes up by a factor of 4. Because the new block
is twice as thick, it has twice as many springs. So the potential
energy goes up by a factor of 8.
Frequency is proportional to sqrt(PE/KE), which is the general version
of omega = sqrt(k/m). So the frequency goes up by sqrt(8/2)=2. The
tone from the thick block is one octave higher.
- 9. Spinning light rod
- The light rod is angled 45 degrees relative to the vertical axis.
The question asked what forces are needed to keep the axis stationary.
Each mass rotates in a horizontal circle. To move in a circle, each
mass needs a centripetal force. This force is supplied by the
person holding the axis, and is transmitted from the axis to the light
When the dumbbell is in the plane of the page (with the top mass going
into the page and the bottom mass coming out of the page), the top
mass needs a force to the left (applied at the top of the axis), and
the bottom mass needs a force to the right (applied at the bottom of