Physics 111 HW 15
DUE Tuesday, 28 June 2016
I07. A 0.5 m long thin stick (m = 2 kg) is glued to two identical hollow spheres (r = 0.1 m, M = 3 kg each) as shown in the diagram at right. It is spun around the axis marked by the circled star. Determine the moment of inertia of this contraption.
T02. The pulley in the figure at right has a radius 0.160 m and a moment of inertia about its axis of rotation of 0.480 kg∙m2. It also has a frictionless bearing. The rope does not slip on the pulley rim. The 4 kg mass is released from rest in the position shown in the diagram. Use forces and torques to find the speed of the 4 kg mass just before it hits the floor. HINT: The tensions in the rope on the left and right sides are no longer the same! How is this so? There is friction between the rope and pulley where it touches the pulley. Thus, although the tension on the left side is bigger than the right, friction between the rope and the pulley makes up the difference.
T03. Two metal disks, one with radius R1 = 2.50 cm and a mass M1 = 0.80 kg and the other with radius R2 = 5.00 cm and mass M2 = 1.60 kg, are welded together and mounted on a frictionless axis through their common center (see figure).
a) What is the total moment of inertia of the two disks?
b) A light string is wrapped around the edge of the smaller disk, and a 2.00 kg block suspended from the free end of the string. If the block is released from rest at a distance of 2.00 m from the floor, what is its speed just before it strikes the floor?
c) Repeat the calculation of part b), this time with the string wrapped around the edge of the larger disk. In which case is the final speed of the block faster? Explain why this is so.
T04. A solid ball is released from rest and goes down a hillside that slopes downward at 65.0o from the horizontal.
a) Find the acceleration of the ball’s center of massand the minimum coefficient of friction needed to prevent the ball from slipping against the ground.
b) Would the coefficient of friction calculated in part a) be sufficient to prevent a hollow ball from slipping? Justify your answer.
c) In part a), why did we use the coefficient of static friction and not the coefficient of kinetic friction?
T06. A 55.0 kg grindstone is a solid disk 0.520 m in diameter. It is mounted on an axle through its center. To sharpen your axe, you push it on the outer edge of the grindstone so that the normal force of the axe on the grindstone is 160 N. The coefficient of friction between the axe and the grindstone is 0.60. The axle of the grindstone is not frictionless; there is a frictional torque between the axle and its bearings of 6.50 N∙m. The stone is operated manually; there is a crank handle that another individual must push tangentially to get the stone spinning. This handle is attached to the stone a distance of 0.50 m from the axle. The stone starts off at rest (not spinning).
a) How much force must someone apply tangentially to the handle to get the stone spinning at a rate of 120 rev/min in 9.00 seconds?
b) After the grindstone attains an angular speed of 120 rev/min, what tangential force on the handle is needed to maintain a constant angular speed of 120 rev/min?
c) How much time does it take the grindstone to come from 120 rev/min to rest if it is acted on by the axle friction alone?
T09. A string is wrapped several times around the rim of a small hoop with radius 0.0800 m and mass 0.180 kg. If the free end of the string is held in place and the hoop is released from rest, calculate:
a) the tension in the string while the hoop descends as the string unwinds;
b) the time it takes the hoop to descend 0.750 m;
c) the angular speed of the rotating hoop after it has descended 0.750 m.
RE01. A solid, uniform ball rolls without slipping up a hill as shown in the figure. At the top of the hill, it is moving horizontally and then it goes over the vertical cliff.
a) How far from the foot of the cliff does the ball land and how fast is it moving translationally just before it lands?
b) Notice that when the ball lands, it has a larger translational speed than when it was at the bottom of the hill. Does this mean that the ball somehow gained energy? Explain.
RE02. A uniform, solid cylinder with mass M and radius 2R rests on a horizontal tabletop (see figure). A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley of mass M and radius R that is mounted on a frictionless axle through its center. A block of mass M is suspended from the free end of the string. The string doesn’t slip over the pulley surface, and the cylinder rolls without slipping on the table top. Find the magnitude of the acceleration of the block after the system is released from rest
a) using torques and forces, and
b) using energy considerations.
RE04. A uniform drawbridge 8.00 m long is attached to the roadway by a frictionless hinge at one end and can be raised by a cable attached to the other end. The bridge is at rest, suspended at 60.0o above the horizontal, when the cable suddenly breaks.
a) Find the angular acceleration of the drawbridge just after the cable breaks.
b) Can you use ωf = ωi + αt to calculate the angular speed of the drawbridge at a later time? Explain.
c) What is the angular speed of the drawbridge as it becomes horizontal?
RE05. An old pulley with friction in its bearing has a mass M = 10 kg and a radius of 0.15 m. A smaller mass (m = 2 kg) is hung from a rope attached to the pulley and dropped from rest. The smaller mass falls 1 m in 2 seconds. Find
a) the frictional heat loss in the bearing (in Joules);
b) the frictional torque in the bearing;
c) the normal force the pulley’s axle exerts on the pulley.
(over)