Lecture 8: Supplementary Notes and Exam-
Below we give a few examples of other oscillators that undergo SHM.
▯ A constrained particle on a spring: A particle of mass m is con-
strained to move along the x axis. It is attached to a light spring whose
other end is ▯xed at point A a distance l from the x axis. The tension
in the spring when the particle is at x = 0 is T. Find the frequency of
motion for small oscillations.
▯ Answer: The tension in the spring increases as it is stretched, but this
turns out not to be important for small oscillations. The new tension
is T + ▯T. The force in the x direction is
▯(T + ▯T)sin▯ = ▯(T + ▯T)(x=l) = ▯Tx=l + H:O:T::
(Assuming x=l ▯ 1, the change in tension is only found in the H.O.T.).
Thus mx = ▯Tx=l which means that the frequency is
! = T=(lm):
1 Note that for a small spring l ▯ resting length of spring, T ▯ kl and so
! = k=m. What would happen if the increase in tension were taken
into account? Would you expect the frequency to increase or decrease?
▯ The buoyancy force B is the upward pressure force on a submerged
object and is always equal in magnitude to the weight of the
displaced by the submerged object. Its magnitude is then
B = ▯ f s;
where ▯ fs the density of the
uid and V is the volume of that part of
the object that is submerged. The buoyancy force B is a property of
uid and is independent of the object it is acting on.
▯ The weight of an object W is of course downward and its magnitude is
W = mg;
where m is the mass of the object. The weight W is a property of the
object and is independent of the
▯ For a
oating object in static equilibrium with a volume V s;esubmerged
below water level, the net vertical force is
B ▯ W = ▯ V g ▯ mg = 0 (at equilibrium.)
e f s;e
▯ For a
oating object displaced from equilibrium, the net force is
B ▯ W = B + (B ▯ B ) ▯ W = ▯ (V ▯ V )g:
e e f s s;e
2 ▯ This net force is nonzero and must be balanced by the acceleration of
the object’s CM. If we denote by z the position of the CM, then
= m g(Vs▯ V )s;e
To get an equation of motion we need to write V ▯Vs s;ein terms of the
CM’s position z.
▯ Worked example: ▯nd the period of oscillation for a 2 cm long cylin-
drical cork bobbing in water about its vertical axis. The speci▯c gravity
of cork is about 0:25.
▯ Solution: For a cyli