Physics 1

Answers are given at the following link.

Prof. Michael Sobel

Data:
G = 6.67 X 10^-11 N-m²/kg²
Gas constant, R = 8.31 J/mole-deg.
Molecular weights: argon, 39.9; neon, 20.2
Radius of earth 6.4 X 10⁶ m; mass of earth = 5.98 X 10²⁴ kg.

Part A: Do THREE of the following four problems.

1. Raindrops are falling vertically on a horizontal roof. Their average speed when they hit the roof is v = 12 m/s. The collision with the roof is completely inelastic, in that the raindrops don’t bounce up at all. There are about 1500 drops per cubic meter, and the mass of each drop is 2.0 X 10^-4 kg.
a) What change in momentum is there in each collision?
b) Suppose the y axis is positive upward. Discuss the sign of the momentum change and of the force on the drop. Explain why there is a downward force on the roof.
c) Given a time Dt, and an area A on the roof, write an expression for the number of drops that hit that area in the time Dt.
d) Find the pressure on the roof. Explain your reasoning briefly.

2. By applying the kinetic theory of gases to a calculation of gas pressure, one deduces the following relation between the average kinetic energy of a molecule (call it “KE”) and the absolute temperature, T:

KE = (3/2)(R/N₀)T,
where N₀ is Avogadro’s number, and R is the gas constant. Use this result in the following problem:
A closed jar contains N molecules of a monatomic gas. Heat Q is added to the gas.
a) Which of the terms in the First Law of Thermodynamics are involved in this process? From the First Law, how is Q related to the change in the average kinetic energy of one molecule?
b) How is Q related to the change in the gas temperature and the molar specific heat?
c) From the above, find the molar specific heat of the gas (as a number).

3. Neon of mass m, at 100 degrees Celsius is added to 20 grams of argon at 0 degrees Celsius. The final temperature is 60 degrees. (In this problem, use the value for specific heat predicted by kinetic theory.)
a) Find the specific heats of argon and neon in J/g-deg. or J/kg-deg.
b) Find m (in grams or kg).

4. A force F = 100 N is used to push a mass of 1.0 kg across a horizontal surface through a distance of 0.5 meters. The velocity of the mass increases from 2.0 m/s to 8.0 m/s.
a) Find the change in macroscopic energy.
b) Find the change in internal energy.
c) Suppose half of this energy goes to the mass and half goes to the surface. The mass has specific heat = 400 J/kg-deg. Find the increase in the temperature of the mass.

Part B: Do FOUR of the following seven problems.

5. In the figure below the masses are as shown, and the coefficient of friction on the surface is 0.55. The pulleys are massless. The system starts from rest and moves until the mass on the right has gone down 0.4 m.
a) Find the force of friction and the work done by friction.
b) Find the change in potential energy.
c) Find the final speed, v.
6. In the figure below, the pulley has radius = 0.12 m and moment of inertia = 0.02 kg-m². The horizontal surface is frictionless. The system starts from rest and moves until the two blocks have final speed = 0.4 m/s. The block on the right has moved down a distance = Dy.
a) Find the angular velocity of the pulley.
b) Find the final kinetic energy of the total system.
c) Find Dy.

7. Mass A (0.2 kg) moves with velocity = 1.5 m/s, and makes an inelastic (this does not mean completely inelastic, i.e. not sticking together) collision with mass B (1.1 kg), which is at rest. Mass A recoils in the direction shown below, with speed = 1.0 m/s.
a) Find the x and y components of the final momentum of A.
b) Find the x and y components of the final momentum of B.
c) Find the final speed of B
d) Find the angle q that the velocity vector of B makes below the x axis.

8. A satellite 1000 kilometers above the earth breaks apart. One fragment (mass = 1.0 kg) is instantaneously at rest. Then it falls down to the earth.
a) Find its initial potential energy, using the exact form for potential energy in the earth’s gravitational field.
b) Find its change of potential energy from when it starts to fall until it hits the earth’s surface.
c) Find its speed just before it hits the ground.

9. Jupiter has a small satellite named Amalthea, which orbits in a circle of radius = 1.8 X 10⁸ m. The period of its orbit is 0.49 days.
a) Find its speed in m/s.
b) Find its acceleration.
c) A new satellite is found at 4.2 X 10⁸ m. Find its acceleration.

10. A physics instructor stands on a frictionless rotating platfrom, holding two 2.5 kg weights at arms length. The weights are 0.6 m from the axis of rotation. The moment of inertia of the instructor-plus-platform is 2.0 kg-m². The system is rotating with an angular velocity of 1.2 rad/s.
a) Find the total angular momentum.
b) He now pulls the weights in, so that they are 0.1 m from the axis. Neglect the mass of the instructor’s arms. Find the final angular velocity.
c) As the instructor pulls the weights in, he does work on the weights. Is this work positive or negative? Explain.

11. The figure below shows a bowl in the form of the interior of a cone. The surface is frictionless, and a small mass m is on it in the position shown. The mass moves in a horizontal circle of radius r = 0.2 m, and with speed = v. F is the vector force exerted on the mass by the bowl; it is perpendicular to the surface.
a) Express the x and y components of F in terms of the magnitude, F, and the angle shown.
b) Write an equation for the y component of Newton’s second law.
c) Write an equation for the x component of Newton’s second law.
d) Eliminate F between these two equations (divide one by the other), and find an equation involving only r, v and q. Find v.