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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.
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.
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.
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.
| ( | ) | ( |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.