Final Examination
Physics 104
19 May 1977

Attention:  in this problem and the next one, some of the given information is not needed to arrive at the correct answer.  You should be able to see which information is pertinent.

#1. One gram-mole of an ideal monatomic gas at pressure p0, volume V0 and temperature T0 undergoes an isothermal expansion, and its volume is doubled.  the change in its internal energy is equal to

a) RT0     b) RT0 log 2    c) zero     d) 3RT0 /2      e) 2RT0

#2.  In problem #1 , if the expansion of the gas is done reversibly, the change in the entropy of the gas plus the environment is equal to

a) R log 2     b) (3 R log 2)/2     c) R     d) zero     e) 2R

#3. An ideal gas undergoes an adiabatic process a ®  b which is shown in the sketch.  If the temperature at a is T0, the temperature at b is

a) T0
b) 4 T0
c) 4(1 -g) T0
d) 3 T0
e) 4g T0

#4.  The following mixture is put into a thermally insulated container
M grams of water at T1 = 30 oC
m grams of ice at T2 = 0 oC

Let

L = heat of fusion of water (per unit gram)
c = specific heat of water (per unit gram)

Suppose that the final equilibrium temperature of the mixture is at temperature T3 > 0 oC. Then the change in the entropy of the system from the time it was first put in the container to the time at which equilibrium is established is equal to

Note: the temperatures in these answers are expressed in oK.

a) m L
b) m L/T2 + mc log T3/T2  +   m c log T3/T1
c) m L/T3 log T1/T2 + M c log T3/T1
d) m c log T2/T3 + Mc logT3/T1
e)  - m L

#5.  A gas consists of a mixture of hydrogen molecules (mass 2.0 amu) and helium atoms (mass 4.0 amu).  If the helium atoms have a root-mean-square speed of 6800 meters/sec, then the root-mean-square speed of the hydrogen molecules is most nearly equal to

a) 6800 b)  4800) c)  13600 d) 3400 e) 9600

#6.  One gram-mole of an ideal monatomic gas at temperature T0 is caused to undergo the cycle shown in the sketch.  The work done by the gas from a to c along the path abc is equal to

a) 3 p0 V0
b) 4 p0 V0
c) 2 p0 V0
d) p0 V0
e) zero

#7. In problem #6, the change in internal energy in going from b to c is equal to
a) zero
b) 2 R T0
c) 4 R T0
d) 6 R T0
e) 3 R T0 / 2

#8. One gram-mole of an ideal gas undergoes the cycle shown in the sketch.  ab and cd are at constant volume, and bc and da are at constant pressure.  If the temperature at a is T0, then the net heat transferred to the system for the full cycle is equal to

a) zero    b) RT0 log 2    c) - RT0 log 2    d) - RT0   e) - RT0 / 2

#9. The linear coefficients of thermal expansion for steel and aluminum are, respectively,
11 ´ 10-6 / 0C for steel
23 ´ 10-6 / 0C for aluminum
A steel rod has a diameter of 3.000 cm at 25 0C; an aluminum ring has an interior diameter of 2.992 cm at 25 0C. The common temperature (in 0C) at which the ring will just slide into the rod is most nearly equal to
a) 83300    b) 250    c) 670   d) 220   e) 80

#10. A standing sound wave is set up in a tube closed at one end as shown in the sketch.  If the speed of sound in air is 330 meters/sec, the next higher resonant frequency (in Hz) that can occur is most nearly equal to
a) 115    b) 880    c) 660   d) 825   e) 413

#11. A wave traveling in a medium is described by the equation
y( x, t ) = (0.10 meters) sin [0.020 x /meters - 25 t / sec]
The phase speed (in meters/sec) is most nearly equal to
a) 0.10    b) 1.50    c) 1250   d) 2.5   e) 4.0

#12. For the same wave as in problem #11, find the value of
y / A
when
x = l / 4 and t = 1 / (8 n)

where A = amplitude; l = wavelength; n = frequency. The answer is most closely equal to
a) zero    b) 0.125    c) 0.250    d) 0.707   e) 1.000

#13.Water (density r) flows in a horizontal pipe with a cross-sectional area A1 and a speed v1.   The pipe turns vertically upwards as shown in the sketch; the vertical part has a smaller cross-sectional area. A2.  The water makes it to a height h where its pressure is equal to atmospheric pressure, p0.  Its gauge pressure in the horizontal pipe is

a) r v12[A12/A22 - 1] + r g h
b) v1A1/A2
c)  r g h
d) p0
e) none of the above

14. A pipe of inner diameter D is coupled with three other pipes, each of inner diameter d.  The flow rates of water in the three pipes of diameter d are 7.0, 6.0, and 4.0 gallons/minute.  If the velocity of flow in the pipe of diameter D is V, then the velocity of the water in the pipe carrying 4.0 gallons/minute is most closely equal to

a) (D/d)2 V    b) 4V/13    c) 13 (D/d)2 V    d) 17 (D/d)2 V    e) 4/17 (D/d)2 V

#15.  A metal ball has a volume V as far as anyone looking at it from the outside is concerned, but it has an unseen cavity inside.  The density of the ball is rM, and the density of water is rW.  When the ball is placed in a pool of water, it sits at the bottom with practically zero force between it and the bottom of the pool.  The volume of the cavity is equal to
a) V rM /rW b) V (rM + rW) /rW    c) V rW /(rM + rW)   d) V / 9.8    e  )none of the above

#16. The law that gives the gravitational force of any particle in the universe on any other particle in the universe,
F = G m1 m2  / r12 2
was discovered and formulated by
a) Archimedes   b) Bernoulli    c) Galileo    d) Kepler    e) Newton

#17.  On a given, spherically symmetric planet, the acceleration of free fall at the surface is g.  The radius of the planet is R. Consider a satellite moving about the planet in a circular orbit of radius 2R.  The angular speed of this satellite (in radians/sec) is equal to
a) Ö[2 g R)]
b)   Ö[2 g / R]
c) Ö [R / 2 g]
d) Ö [g / (8 R)]
e) none of the above

#18.  A simple pendulum of length L and on planet earth has a period T.  On the surface of another planet whose gravitational free fall acceleration is 4.9 meters/sec2, the period of the same simple pendulum would be

a) T / 2
b) T / Ö2
c)
d) 2 T
e) Ö2 T

#19. A mass at the end of a vertical spring  oscillates up and down in accordance with the equation

x = ( 0.01 meter ) sin ( t / 5 sec)

If the total mechanical energy of the oscillating mass is 10-6 joules, then its mass (in kg) is most nearly equal to

a) 0.33    b) 0.50    c) 1.00    d) 2.00    e) 3.00

#20.  A uniform beam of mass M and length L is supported at its ends by two light vertical cords so that it makes an angle q with one of the cords, as shown in the sketch.  The moment of inertia of a uniform beam about an axis through its center and perpendicular to the beam is ML2 / 12.  Someone cuts the right-hand cord.  The angular acceleration of the base immediately afterwards is given by

a) Ö(2 g L)    b) g tan q / L    c)   3 g sin q / (2L)     d) Ö(6 g L)     e)  none of the above

#21. A uniform disk is rolling along the ground with a constant speed v0.  It encounters a plane inclined at an angle q with respect to the horizontal.  The disk rolls without slipping up the plane.  the maximum distance, L, up the plane that the disk will travel is most nearly equal to

Note:  Moment of inertia of a uniform disk about its central axis is MR2/2, where R is the radius of the disk.
a) Ö(2 g L) cos q    b) v0.2/(2g sin q)    c) 3 v0.2/(4g sin q)    d) 3 v0.2/(4g cos q)     e) v02 sin q /(2g)

#22. The velocity of a particle of mass m = 9.31 ´ 10-31 kg is v = (3 ´ 107 meters/sec ) i at a given moment of time when its position has cartesian coordinates
x = 3 ´ 10-10 meters; y = 4 ´ 10-10 meters; z = 5 ´ 10-10 meters
At that moment of time, the angular momentum of the particle about the origin is
a) 7 ´ 10-10  meters    b) 27 i  ´ 10-24 kg m/sec    c) ( 135  j - 108 k ) ´ 10-24 joule-sec
d)16.2 ´ 10-17 joules    e) ( -9  i + 12 j ) ´ 10-3 kg m2 / sec

#23. Dick and Jane, a typical suburban couple, are in outer space (in space suits) at opposite ends of a light, strong cord of length 10 meters.  Dick, together with his suit, has a mass of 175 kg, and Jane, together with her suit, has a mass of 100 kg.  they are rotating relative to the stars with an angular speed of 5 radians/sec.  Dick and Jane each pull on the rope and thereby draw closer together until they are only 5 meters apart.  Their final angular speed (in radians/sec) is
Note:  The solution to this problem can be found shortly and simply.