PHY 102 Problem Set 1
1. Consider a simple pendulum, with a mass m swinging at the end of a
light rigid rod of length l. The pendulum swings with a displacement amplitude
a.If its starting point form rest is
(a) x = a
(b) x = −a
find the different values of the phase constant φ for the solutions
x = a sin(ωt + φ)
x = a cos(ωt + φ)
x = a sin(ωt − φ)
x = a cos(ωt − φ)
For each of the different values of φ, find the value of ωt at which the pendulum swings through the positions
√
x = a/ 2
x = a/2
and x = 0
for the first time after release from x = ±a
2. Show that the values of ω 2 for the three simple harmonic oscillations
(a),(b),(c)in the diagram are in the ration 1:2:4.
3. The displacement of a simple harmonic oscillator is given by
x = a sin(ωt + φ)
If the oscillations started at time t = 0 from a position x0 with a velocity ẋ = v0
show that
(a) tan φ = ωx0 /v0
(b) a = (x20 + v02 /ω 2 )1/2
4. The general form of the energy of a simple harmonic oscillator is
1
E = 12 mass (velocity)2 +
1
2
stiffness (displacement)2
Set up the energy equation for the oscillators in figures (a), (b), (c), (d), (e) and
(f) and use the expression
dE
dt
=0
to derive the equation of motion in each case.
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5. Two simple harmonic motions of the same frequency vibrate in directions
perpendicular to each other along the x and y axes. A phase difference
δ = φ2 − φ1
exists between them such that the principal axes of the resulting elliptical trace
are inclined at an angle to the x and y axes.Show that the measurement of two
separate values of x (or y) is suffiecient to determine the phase difference.
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