Solutions to Conceptual Questions from Homework 1

Physics 224 Homework 1 (solutions) (2006 Spr)
Solutions to Conceptual Questions from
Homework 1
Young, Q13.1: An object is moving in SHM of amplitude A on the end of a spring. If the amplitude is
doubled, what happens to the total distance the object travels in one period? What happens to the period? What
happens to the maximum speed of the object? Discuss how these answers are related.
A Brief Answer:
From the definition of Amplitude, the body will travel a total distance of 4A in one period (eg.
from +A to –A then back to ++A). In SHM, the period is independent of Amplitude, thus the period
remains the same. From considering the equation describing energy conservation in each case
separately, one can easily show that the total energy E in the second system is 4 times that in the
first. Thus the max. velocity (when the potential energy U=zero and hence E=1/2 kv 2 ) is 2 times
the max. velocity of the first system. Basically the distance traveled doubles, but the speed at all
points in the cycle doubles such that the time taken for 1 cycle remains the same.
Young, Q13.2: Think of several examples in everyday life of motions that are at least, approximately
simple harmonic. In what respect does each differ from SHM?
A Brief Answer:
Swings, Pistons in a car engine, pendulums in grandfather clocks, watch springs, rocking horses,
water in a U-tube, etc etc
Young, Q13.8: You are captured by aliens, taken into their ship, drugged. You awake some time later and
find yourself looked in a small room with no windows. All they have left you with is your digital watch, your
school ring, and your long silver necklace. Explain how you can determine whether or not you are still on Earth, or
if you have been transported to Mars,
A Brief Answer:
Using your necklace and ring, you can make a pendulum. You can then set this pendulum swinging
with SHM (ie. With small amplitude), and use your digital watch to measure the period T n of the
oscillations. (Measuring the time for (say) 10 oscillations, then dividing by 10 will increae the
accuracy of the measurement of course.) You can also estimate the length L n of the pendulum to a
fair degree of accuracy. If you are still on Earth, then you expect
where
Hopefully it will also have occurred to you that you should also check whether you really can
distinguish between the period expected on Earth versus that expected on Mars…. (cont)
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Physics 224 Homework 1 (solutions) (2006 Spr)
Where is the PHYSICS behind the situation on Earth and that on Mars? … From the above
equation, surely it can only be in the Gravitation acceleration experienced by the make-shift
pendulum.
What is the PHYSICS behind the Gravitational acceleration? … Newtons law of Gravitation, and
that fact that the Mass and Radius of Mars is different to those of the Earth.
You should recall that for a body of mass M, and radius R (ie a planet), then the accceleration due
to gravity at the surface is
[See Young Sect.12.2 for a review]
Given the mass and radius of Mars, the acceleration on the surface is actually 3.7 m s -2 . Thus it
should be relatively easy to measure a deviation between the Period of the necklace pendulum
expected on Earth and that on Mars.
Young, Q13.13: If a pendulum clock is taken to a mountain top, does it gain or lose time, assuming it is
correct at lower elevations? Explain your answer.
A Brief Answer:
Again, as in Q13.8, for a pendulum in SHM, the period is given by
where
where the R.H.S. is expressed in terms of the Mass and Radius of the Earth, and the height (h) of
the mountain. Obviously the higher the mountain, the lower the value of the ‘local g’ and hence the
longer the period. Thus the clock will “lose time” (eg. if the period is 1 oscillation (or ‘tick’) per
second at the lower evelation, it will obviously take 60s to make 60 ticks. However given its lower
period, after 1 real minute it will have made (slightly) less than 60 ticks at the higher elevation,
and hence will appear to run slow.)
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