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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 v.

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 9 metres.
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 ratio.

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 flat graph. 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. (A limiting 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 block.

The kinetic energy is easy: The mass doubled so the kinetic energy doubled.

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 rod.

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 the axis).