Physics 101 - Final Exam 1996

Lecturer: C.F. Perdrisat

A block of mass M=4 kg resting on a 30° incline is attached to a second block of mass m by a massless cord that passes over a frictionless peg as shown. The static coefficient of friction between the block and the incline is _s = 0.4.

1. find the minimum and maximum values of m, for which the system will be in static equilibrium. Hint: these extremes correspond to the frictional force being maximum, in either one of the possible directions of motion!
2. if m=1 kg, the system will be in static equilibrium; what is then the frictional force acting on the 4-kg block, and what is its direction?

A first block of mass M is attached to the end of a very long rope hung over a peg at a distance 2d from the wall; the other end is attached to the wall at the same height as the peg. A second block of mass m is attached to the middle of the string, half way between the wall and the peg (see picture). It follows that the included angles between the rope and the horizontal at the wall and at the peg have a magnitude . Isolating the forces acting on m and M, find at what vertical distance y_o the system is in equilibrium.

A homogeneous cylinder of mass M=100 kg and radius R=0.3 m is mounted such that it turns without friction and clockwise around its fixed symmetry axis. It is rotated by a drive belt that wraps around its perimeter and exerts a constant torque (see figure). At time t=0, its angular velocity is zero. At time t=30 s, its angular velocity is 10 rev/s. For a cylinder I= ½ MR².

1. what is its angular momentum L₀ at t=30 s?
2. at what rate is the angular momentum increasing for t<30 s?
3. what is the torque acting on the cylinder for t<30 s?

A uniform boom of length L=5 m and total mass M=150 kg is connected to the ground by a hinge at the bottom, makes an angle 45° with the horizontal and is supported by a horizontal cable attached at the top as shown.

1. what is the tension T in the cable?
2. if the cable is cut, what is the angular acceleration of the boom at the instant the cable is cut? The moment of inertia of a rod rotating around an axis perpendicular to its length axis at one end is  I = (1/12)ML² + M(L/2)².

The mass of Saturn is M = 5.69x10²⁶ kg

1. find the period of its moon Mimas, whose mean orbital radius is 186000 km.
2. find the mean orbital radius of its moon Titan, whose period is 1.38x10⁶s.

The universal gravitation constant is G = 6.67x10^-11N·m²/kg².

A mass M=2 kg oscillates horizontally with an initial amplitude A₀=3 cm on a spring of force constant k=400 N/m. The damping is weak!

1. what is the period?
2. what is the total initial energy, E_tot0?
3. if the energy decreases by 1 percent per period, find the damping constant, b, and the Q factor.
4. how much time will it take for the energy in this spring-mass oscillator to decrease to 1% of its initial value E_tot0?

A typical loud sound wave with a frequency of 1 kHz has a pressure amplitude of 10^-4 atm (1 atm ~10⁵ Pa and _air = 1.29 kg m^-3). Use the value v=343 m/s for the speed of sound propagation in air at 200° C.

1. what is the wavelength of this sound?
2. at t=0 the pressure is a maximum at some point x₁. What is the displacement at that point at t=0?
3. what is the maximum value of the displacement at any time and place?   (Note: this material was not covered in 2000 - so ignore this part)