Download 18 /9 - University of St. Thomas

Physics 111 HW 18
Due Thursday, 7 July 2016
/9
PP01. A 1.80 kg monkey wrench is pivoted 0.250 m from its center of mass and allowed
to swing as a physical pendulum. The period for small-angle oscillations is 0.940
s.
a) What is the moment of inertia of the wrench about an axis through the pivot?
b) If the wrench is initially displaced 0.400 radians from its equilibrium position, what is the angular
speed of the wrench as it passes through the equilibrium position?
PP02. Two identical, thin rods, each with mass m and length L, are
joined at right angles to form an L-shaped object. This object is
balanced on top of a sharp edge as shown in the figure at right. If
the L-shaped object is deflected slightly, it oscillates. Find the
frequency of oscillation.
L
LL
D01. An unhappy 0.300 kg mouse, moving on the end of a spring with
force constant k = 2.50 N/m is acted on by a damping force Fx = – bvx.
a) If the constant b has the value 0.900 kg/s, what is the frequency of oscillation of the mouse?
b) For what value of b would give a period of oscillation that is twice that of the period with no damping
at all?
c) For what value of b will the motion be critically damped?
D03. A 250-g mass is mounted on a spring of constant k = 3.3 N/m. The damping constant for this system
is b = 8.4  10-3 kg/s. How many oscillations will the system undergo during the time the amplitude
decays to 1/e of its original value?
SHM01. A mass m is mounted between two springs of constants k1 and k2 as shown in the figure. When
the mass is in the equilibrium position for one spring, it is also in the equilibrium position of the other.
Determine the frequency (in Hz) of oscillations of this mass in terms of k1, k2, and m. You may use
forces or energy, but you should know how to do both!
k2
k1
m
SHM03A. A solid cylinder of mass M and radius R is
k
mounted on an axle though its center. The axle is
attached to a horizontal spring of constant k, and the
M
=
cylinder rolls back and forth without slipping (see figure).
Determine the frequency (in Hz) of this motion in terms of k, M, and R. You can get it straightforwardly
by using forces and torques to find the effective spring constant, but you may also use energy
considerations. Note that since the angular acceleration is not constant, the torque friction exerts is not
constant, which means that µs is not constant either. Solve the torque equation for µs and plug this into
the Fx equation to find the angular frequency.
(over)
SHM05. An 2cm by 2cm by 2cm ice cube has a mass of 7.36 g. It is placed in a glass of water.
When it just floats there, the weight of it is exactly balanced by the buoyancy force. If you
push it down a distance x, though, it bobs up and down with simple harmonic motion. The net
force on it in this situation is - A x, where  is the density of water (1000 kg/m3), A is the
cross-sectional area of the cube, and x is how far the cube is above or below the equilibrium
position. Determine the frequency of oscillation of the cube.
SHM06. A thin rod with length L = 2.00 m
and mass m = 1.00 kg hangs vertically from

a hinge at its top end. (marked with a circled
star in the diagram). A mudball (essentially
m = 1.00 kg
a point mass of 1.00 kg) flies horizontally
L = 2.00 m
toward the bottom end of the rod with a
speed of 0.40 m/s. When it hits the rod, it
sticks to it. Assume the hinge is frictionless m = 1.00 kg
and there is no damping.
v = 0.2 m/s
a) Find the moment of inertia I of the
rod/mudball system after the mudball has stuck to the rod.
b) Find the frequency of the resulting oscillations. You may express your answer in rad/s or Hz (your
choice).
c) Find the maximum angle the rod makes with the vertical after the collision. (CAREFUL: The
angular frequency ωo of the oscillating rod is NOT the same as the angular velocity ω of the rod
right after the collision!
Angular frequency ωo is a constant that gives an idea how long it takes to go back and forth once.
Angular velocity ω varies depending where the pendulum is in its motion. It is greatest when the
pendulum is vertical, and zero at θmax.)
SHM08. An ideal (negligible mass) spring with k1 = 60 N/m has one end fixed to the wall and the other
end attached to a block of mass m = 1.5 kg. The other side of this mass is attached to a string which is
wound once around a pulley mounted on the floor (m = 1.5 kg, R = 0.050 m) and is attached to another
mass (m = 1.5 kg). This mass, in turn, is attached to an ideal spring with k2 = 40 N/m. When the
masses are at rest, both springs are at their equilibrium positions. The right mass is displaced 0.10 m
and released. Determine the frequency of the resulting oscillations in rad/s. There is no friction
between the two masses and the ground. Do this using torques/forces AND energy considerations.
m = 1.50 kg
k1 = 60 N/m
m = 1.50 kg
R = 0.050 m
(over)
m = 1.50 kg
k2 = 40 N/m