Homework Ch. 10 Rotations - Due M 4/23 in class.
PHY 161 General Physics I: Mechanics and Thermodynamics
Physics Department --- Mercer University --- Spring 2007
PLEASE NOTE: The instructor’s preference is that your homework solutions be handwritten on printed
copies of these pages and/or blank standard printer paper sheets (8 ½” x 11”), front and back... But if
absolutely necessary, you may use notebook pages; in that case, please remove the paper fringes.
1. A massless rectangular grid has four masses at its corners: m1 = 1 kg, m2 = 2 kg, m3 = 3 kg,
m4 = 4 kg. The sides have lengths a = 3 m and b = 4 m. There is a force acting on each mass.
Their magnitudes are F1 = 10 N, F2 = 20 N, F3 = 30 N, F4 = unknown. The forces act in the
plane of the page and have angles θ1 = 30°, θ2 = 40°, θ3 = 20°, θ4 = unknown.
a) If the net force on the grid is zero, find the force on mass #4 (F4, θ4).
b) Calculate the moment of inertia I of the grid about an axis passing through mass #4
perpendicular to the page. For the same axis, calculate the net torque τ provided by the
forces, and the resulting angular acceleration α. Are τ and α clockwise or counter-clockwise?
c) Repeat the calculation for an axis passing through the exact geometric center of the grid
perpendicular to the page: find I', τ', α' and whether clockwise or counter-clockwise.
2. A pulley with radius R = 0.15 m and mass m = 1.2 kg has two masses hanging by a rope on
its right and left sides: mR = 5 kg, mL = 3 kg. The pulley’s moment of inertia is I = ½ mR2.
Use Newton’s First Law on the pulley and hanging masses to produce three equations
relating the masses (m, mR , mL), the tensions on the two ropes (TR, TL) and the linear
acceleration (a, upward) of the left mass. Solve these to predict the value of the acceleration.
EXPERIMENTAL RESULT: When the masses are actually released from rest, the right one
drops by 2m in 1s. Find the true acceleration aexp. We account for this result by assuming that
the moment of inertia is actually I = CmR2, with C ≠ ½. Find the true value of C.
3. Two wheels are free to rotate around fixed axes but they are connected by a massless belt.
This belt wraps around them tightly and will not slip no matter how they move. The large
wheel has radius R = 80 cm, mass M = 20.5 kg and moment of inertia IR = ½ MR2. The small
wheel has radius r = 5 cm, mass m = 80 g and moment of inertia Ir = ½ mr2. The wheels and
belt begin at rest. Then a constant force F = 3 N is applied along the rim of the larger wheel
for a duration Δt = 10s. As a result the wheels start rotating, and there is a difference in the
tensions on the bottom (T) and top (T’) parts of the belt.
a) Which part of the belt is at the higher tension, and what is the difference in the tensions?
(Note: You cannot find the actual tensions themselves as they depend on how tightly the belt
is wrapped around the wheels. Only the difference affects the motion of the wheels.)
b) What are the angular accelerations of the wheels, αR and αr?
c) What are the final angular velocities of the wheels, ωR and ωr?
d) What is the final total kinetic energy K?
e) Find the total angle of rotation of the large wheel θR. Use this to verify that K = WF, where
the work done by the force is WF = FΔs = FRθR.