HW-3
1. A body of uniform cross-sectional area A and mass density of ρ floats in a liquid of
density ρ0 and at equilibrium displaces a volume V. Show that the period of small
oscillations about the equilibrium position is given by
τ = 2π V / gA
where g is the gravitational field strength.
2. A pendulum is suspended from the cusp of a cycloid cut in a rigid support (see
Figure). The path described by the pendulum bob is cycloidal and is giben by
x = a (φ − sin φ ) , y = a (cos φ − 1)
where he length of the pendulum is l=4a, and where φ is the angle of rotation of the
circle generating the cycloid. Show that the oscillations are exactly isochronous with
a frequency ω = g / l , independent of the amplitude.
y
2a
0
l
m
x
2a
Suggestion: write the total energy and show that, with an appropriate choice of
φ
variable of u ≡ 4a cos , the energy can be put into the standard form of a simple
2
harmonic oscillator.
3. A particle of mass m is at rest at the end of the spring (force constant k) hanging from
a fixed support. At t=0, a constant downward force F is applied to the mass and acts
for a time t0. Show that after the force is removed, the displacement of the mass from
its equilibrium position (x=x0, where x is down) is
F
x − x 0 = [cos ω o (t − t o ) − cos ω o t ]
k
2
where ω 0 = k / m .
4. The support of a simple pendulum of natural frequency ωo is moved horizontally with
the simple harmonic motion Asinωt where ω ≠ ωo. With small amplitudes, calculate
the steady-state motion of the pendulum bob. Show that the pendulum appears to
pivot about a point for which the natural frequency would equal ω.
5. Consider the mechanical system shown below with a dash-pot (the dash-pot produces
a friction force proportional to the moving speed), a spring and a mass attached end to
end. The mass is subjected to a force F=Fosinωt (in addition to the spring k).
Calculate the amplitude of the motion of the mass m.
k
m
F