Download FIFTH EXAM -- REVIEW PROBLEMS

Physics 2210
Spring 2009
George W illiams
FIFTH EXAM -- REVIEW PROBLEMS
Table 10-2 in your text is assumed, and will be provided with the exam.
1.
W e approximate an aircraft propeller with the shape shown.
If it is assumed to have uniform density, and the thickness
increasing as t = A ,x ,, calculate the mass moment of inertia
for rotation about the y axis. NOTE: All curves have the
same shape.
2.
Assume that the planet Uranus has a radius of 2.50 × 10 7 m. A moon orbits the planet with a period of 30.0
days at a distance of 7.50 × 10 7 m above the surface of the planet.
3.
(a)
(b)
(c)
Calculate the mass of Uranus from this data.
Find g on the surface of Uranus.
Calculate the average density of Uranus.
(a)
The mass of Jupiter is 1.88 × 10 27 kg. Its satellite, Io, has an orbital period of 42 hours. Find the
radius of Io's orbit.
Given g on the surface of the moon (.167 m/s 2) and a radius of 1.74 × 10 3 km, find the average density
of the moon.
Find the value of g 2000 miles above the surface of the moon.
A satellite is in orbit above the Earth. If the mass of the Earth were suddenly doubled, find the ratio
V new/V old, giving the change in velocity of the satellite necessary to keep the orbit radius the same.
(b)
(c)
(d)
4.
A certain hypothetical planet has a radius of 500 km and a surface gravity of 3.0 m/s 2.
(a)
(b)
W hat is the gravitational acceleration of an object 100 km above the surface of the planet?
W hat is the mass of the planet?
5.
In the drawing shown, M 1 and M 2 are stars and M 3 is a planet. If
M 1 = 6.0 × 10 29 kg, M 2 = 3.0 × 10 27 kg and M 3 = 4.0 × 10 22 g, find the
gravitational force (magnitude and direction ) on M 3. Take a = 8.0 × 10 7
km.
6.
Jupiter has a moon Io, whose period about the planet is 42.1 hours.
(a)
(b)
7.
If the orbit of Io has a radius of 4.22 × 10 8 m, find the mass of Jupiter.
Derive Kepler's Third Law starting with Newton's Law of Gravity and the basic mechanics of
circular motion.
The Earth circles the Sun in an orbit whose radius (take the orbit as circular) is 92,000,000 miles, in a
time of 365 days. The diameter of the Sun is 860,000 miles. Find the mass and the average density of
the Sun. (Be sure to show how you got these numbers.)
8.
An astrophysicist is making a model of a star. He takes the mass as M o and the radius as R o. He
proposes that the density vary as D(R) = Do(1 - "R 2).
(a)
(b)
Find an expression for " in terms of M o, R o and Do.
If a planet orbits this star at a distance of 100,000,000 miles with a period of 88 days, find the
mass (in kg) of the star.
9.
The radius of the Earth is 4000 miles. Take a model of the Earth in which an inner sphere of radius
2000 miles has an average density of 9.0 × 10 3 kg/m 3. Find the average density of the outer part of the
Earth such that g on the surface remains 9.80 m/s 2.
10.
Consider a planet where the density varies according to the relationship D = Do(1 - "R), where Do is the
density at the center, " is a constant, and R is the distance from the center. The radius of the planet is
R o.
(a)
(b)
11.
A satellite of mass of 150 kg is in a circular orbit 8000 mi above the Earth's surface.
(a)
(b)
(c)
(d)
12.
(a)
(b)
(c)
(d)
13.
Calculate the velocity of the satellite in this orbit.
Calculate the total energy of the satellite in this orbit.
Calculate the minimum amount by which the satellite's velocity must be increased in order that it
escapes from the Earth. Ignore the Sun and all other planets.
Assume that a black hole has a total mass 3.00 times the mass of the sun. Calculate the speed of a
small object in a circular orbit 1000 miles from the center of the black hole.
Calculate the speed (in m/s) of a satellite in a circular orbit 1000 mi above the surface of the
Earth.
Calculate the total energy of the satellite in (a) if its mass is 1500 kg.
Calculate the gravitational force between two elephants (assumed to be spheres) if there is 6.00 m
between their centers. Take the mass of each elephant as 4000 kg.
Assume that a black hole has a total mass 3.00 times the mass of the Sun. Calculate the speed of
a small object in a circular orbit 1000 miles from the center of the black hole.
The planet Pluto has a moon, Charon, whose orbital period is 6 days. Assume that the mass of Charon
is small compared with the mass of Pluto (probably not true). Assume that the mass of Pluto is 0.10
M E.
(a)
(b)
(c)
14.
Calculate the mass of the planet.
Calculate the value of at a point below the surface of the planet, which is a distance R o/3 from
the center of the planet.
Find the radius of Charon's orbit.
If the radius of Pluto is 200 km, find the escape speed from its surface.
Assume that Pluto is in a circular orbit of radius 5.90 × 10 9 km about the Sun. Calculate the
orbital period of Pluto about the Sun.
Assume that a geologist wishes to make a model of the Earth where the density varies as
D(R) = Do(1 - "R 2). Assume that the radius of the Earth (R E) and the mass of the Earth (M E) are as given
on the data sheet. Use Do = 7.50 × 10 3 kg/m 3.
(a)
(b)
Find the value of ".
Calculate the value of g at point R = R E/2.
If you can't do (a), leave " in (b) as a symbol.
15.
Captain Kirk and the Enterprise are in orbit about a planet whose radius they measure as 8250 km.
W hen they are in a circular orbit a distance of 24,000 km above the surface of the planet, they measure
the period of that orbit as 3.34 hrs. Calculate the density of the planet.
16.
The density of a star is approximated by D = Do(1 - "R 2), where " and Do are constants, and the density
goes to zero at the surface, R = R o. The radius of the star is 1.20 × 10 6 km, and its total mass is
8.00 × 10 30 kg.
(a)
(b)
(c)
17.
The Starship Enterprise is in a circular orbit about an unknown planet. The kinetic energy of the
Enterprise is 1.00 × 10 19 J and its mass is 4.00 × 10 7 kg. The orbit is 14.0 × 10 6 m above the surface of
the planet, whose radius is 7.00 × 10 6 m.
(a)
(b)
(c)
18.
(b)
22.
Find the total energy (kinetic plus potential) of the earth in its orbit (assume a circular orbit).
Calculate the total speed needed for the earth to leave its orbit and escape from the sun. That is,
find the velocity at its present distance from the sun needed to escape.
Assume that a satellite orbits the earth in a circular orbit 20,000 miles above the earth's surface in
exactly 24.0 hours (not necessarily true values). Assume R E = 4000 mi.
(a)
(b)
(c)
Calculate the mass of the earth from this data.
Calculate the density of the earth in kg/m 3 from this data.
If the satellite has a mass of 1.00 × 10 4 kg, calculate its total energy in this orbit.
(a)
If the mass of the earth were doubled and its radius tripled, what would be the new weight of a
person who now weighs 200 pounds?
The period of Halley's comet about the sun is 76.0 years. Calculate the value for the semi-major
axis of its elliptical orbit.
Calculate "g" 1000 km above the surface of the moon.
Calculate the period of a satellite in a circular orbit about the moon 1.0 km above the surface of
the moon.
(b)
(c)
(d)
23.
Calculate the mass of the sun in kg using the above numbers.
If the radius of the sun is 500,000 miles, find its mass density in kg/m 3 using the mass obtained in
part (a).
For the following question, use the data from the data sheet and an earth-sun distance of 93,000,000
miles between their centers.
(a)
(b)
21.
If he chooses Do = 1.00 × 10 4 kg/m 3 find the magnitude and units of " that gives the correct mass
for the earth.
Using the above form for the mass density, calculate the magnitude of the effective "g" at
R = 4.00 × 10 3 km from the center of the earth.
The planet Mercury orbits the sun in a period of 100.0 days at a distance of 40,000,000 miles (number
are not quite the true value).
(a)
(b)
20.
Calculate the potential energy (numerical value) of the Enterprise in this orbit. Ignore influences
other than this planet.
Determine the total energy (numerical value) of the Enterprise.
Find the average mass density of this planet.
A geophysicist decides to model the earth as having a mass density that diminishes linearly with the
distance from the center using D(r) = Do ! "r where Do and " are constants and r is the distance from the
center of the earth.
(a)
19.
Calculate ".
Determine Do.
Find the value of g at point R o/3 (inside the star).
Planet Mercury orbits the sun in a circular orbit in 95.0 days at a distance of 6.00 × 10 7 km from the
center of the sun (not the true values).
(a)
(b)
(c)
Find the mass of the sun from the above data. (Do not use the data sheet.)
If the average density of the sun is 1,400 kg/m 3, what is the sun's radius?
Calculate the speed of Mercury in its orbit.
24.
You make a model of the earth with the density varying as D = Do(1 - "R 2), with
Do = 8000 kg/m 3. Given that the mass of the earth is 6.00 × 10 24 kg and the radius is
6.40 × 10 6 m, calculate:
(a)
(b)
25.
the value of ",
the value of "g" at a point, P, 1/3 of the distance from the center of the earth to
the surface. If you failed to get a numerical answer (a), leave " as a symbol in
your solution to (b).
A 1.75 kg mass is supposred by a spring whose spring constant is
170 N/m. At the equilibrium position (with the mass on the spring) the
mass is given an initial velocity of 0.155 m/s upwards.
(a)
(b)
(c)
(d)
Calculate the angular frequency, T, of the resulting oscillation.
Calculate the frequency of the resulting oscillation.
Calculate the amplitude of the motion.
After 342 seconds the amplitude is observed to be 0.500 cm. W hat
percent of the original energy of oscillation then remains?
26.
A block of mass 0.5 kg sits on frictionless table. It is attached to a wall
with a spring whose spring constant is 5.0 N/m. Initially the spring is
neither stretched nor compressed. At t = 0 the mass is given a velocity of
0.5 m/s to the right.
(a)
(b)
(c)
27.
W hen one end of a wire of given length and composition is twisted through a small angle 2 relative to
the other end, it exerts a restoring torque proportional to this angle, i.e., J = -61. A torsion pendulum
consists of such a wire, upper end fixed, with a weightless crossbar at the bottom to which two equal
masses are attached, each of mass M and each a distance r from the wire.
(a)
(b)
28.
Find the maximum deflection from the equilibrium position.
Find the maximum potential energy of the system.
Find the angular frequency, the frequency, and the period of the
oscillation.
W rite the differential equation that governs the oscillations of the "dumbbell" in the horizontal
plane.
If M = 0.500 kg, r = 0.200, and 6 = 12.0 newton Ameter/radian, what is the period for small
oscillations?
A torsion pendulum is constructed consisting of two spheres whose
centers are 10.0 cm apart. Each sphere has a mass of 17.5 g. The
spheres are small enough that they can be considered point masses,
and the rod connecting them has negligible mass. Oscillation in the
horizontal plane is initiated with an initial angular speed of 0.0275
rad/s, and an initial angle of 2o = 0°. The maximum displacement is
observed to be 0.325 radians.
(a)
(b)
(c)
Calculate the natural angular frequency of oscillation.
Calculate the total energy in the system.
Calculate the period.
29.
W hen a mass m 1 is attached to the spring the equilibrium position is y 1 =
12.50 cm below the support. The spring constant is 215 N/m. Now a
mass m 2 is attached and the system released from rest at y 1. (All
coordinates are measured to the same point on m 1.)
(a)
(b)
(c)
Calculate the period of the resulting oscillation.
Calculate the maximum downward displacement measured from
the support.
Calculate the energy in the oscillation only.
30.
A massless beam with a mass m on its end is pivoted at
its left end and supported by two Hooke's law springs as
shown. Initially spring 1 is extended from equilibrium
and spring 2 is compressed.
(a)
(b)
31.
Given a mass on frictionless surface oscillating according to the following function (distances are in
meters, frequency in rad/s): x = 3.25 cos (0.435t + 62.5°).
(a)
(b)
(c)
(d)
(e)
32.
Calculate the frequency, f, of the oscillations.
Calculate the period of oscillations.
Calculate the amplitude of the oscillations.
Calculate the maximum velocity.
If the mass is 25.0 grams, calculate the energy stored in the system.
Given a mass on a frictionless surface attached to the wall by a Hooke's Law spring of constant k = 375
N/m. The mass m has a value of 1.25 kg. A bullet of mass 3.00 grams and velocity 250 m/s, strikes the
mass m and sticks to it at t = 0. Initially, the spring is not compressed and the mass is at x = 0.
(a)
(b)
(c)
35.
Find how far below the initial position the body descends.
Calculate the frequency.
Find the amplitude of the resulting motion, which is assumed to be simple harmonic.
The position of a mass on a spring undergoing simple harmonic motion is given by x = (42.5 × 10 -3) cos
20t - (37.5 × 10 -3) sin 20 t. Distance are in meters, time in seconds, and all other quantities have the
usual and appropriate SI units.
(a)
(b)
(c)
(d)
(e)
34.
Calculate the velocity at t = 0.
Calculate the position at t = 0.
If the mass is 0.177 kg, calculate the energy in the oscillations.
Express the solution above as A cos Tt + B sin Tt. Calculate A and B.
How far does the mass travel in 3.00 s?
A massless spring of force constant 19.0 N/meter hangs vertically. A body of mass 0.20 kg is attached
to its free end and then released. Assume that the spring was unstretched before the body was released.
(a)
(b)
(c)
33.
For a small amplitude of oscillation, calculate the
angular frequency of oscillation.
Now include the effects of the beam by giving the
beam a mass M and a moment of inertia of 1/3
ML 2, for rotation about the pivot. Calculate the
new angular frequency of oscillation.
Calculate the frequency, f, of the resulting oscillation.
Calculate the amplitude of the oscillation.
Calculate the energy stored in the oscillations.
A simple pendulum with a length of 2.15 m, is oscillating with a
small amplitude. Initially the angle 2 = 0.050 rad and the initial
velocity along the arc is ds/dt = !0.42 m/s. (The positive direction is
counterclockwise around P.)
(a)
(b)
(c)
(d)
Calculate the amplitude of the motion, in radians.
Calculate the maximum value of ds/dt.
W rite a complete solution in the form 2 = D cos ( Tt - N), with
D, T and N numerically evaluated, with units and signs.
W rite a complete solution in the form 2 = A sin T + B cos Tt,
with A, B and T numerically evaluated, with units and signs.
36.
37.
At t = 0, the system shown is given an initial velocity of ! 1.45 m/s.
At t = 0 its position is x = + 0.0740 m. The spring constant is 175
N/m, and the mass is 0.25 kg. The spring is a Hooke's Law spring.
(a)
(b)
(c)
(d)
38.
Calculate the amplitude of the motion.
Calculate the total energy in the oscillations.
Calculate the velocity at x = 0.
W rite a complete solution for the motion in the form
x = A cos Tt + B sin Tt with A, B and T numerically evaluated.
An object of mass M on a frictionless surface, is started at t = 0 with a velocity to the left in the drawing
of magnitude of 0.370 m/s, and position of x = +3.20 cm. M = 1.25 kg; k = 47.5 N/m.
(a)
(b)
(c)
39.
A physical pendulum in constructed consisting of a disc with three
smaller discs attached as shown in the drawing. The system is pivoted
at P. The smaller discs are attached on the rim of the large disc at 90°,
180° and 270° from P. The mass of the large disc is 2.75 kg, and its
radius is 0.300 m. The small discs have masses of 0.65 kg and radii of
0.090 m. Calculate the frequency of small angle oscillations in Hz.
W rite a complete solution for the motion in the form
x = A sin( Tt ! N), with all constants numerically
evaluated, and with proper units. [Note: There m ay be
m ore than one possible solution for the phase angle. Be
sure the one you choose does what it is supposed to do.]
Calculate the total energy of the system.
If the total energy is has diminished to exactly 1/2 of its
initial value in 5.00 minutes (300 s), calculate the value of
b in the equation (see data sheet for solution):
A uniform solid sphere is attached to a very light rod of length L.
The rod is attached to a pivot at P on one end, and to the sphere on
the other end.
(a)
(b)
Calculate the angular frequency for motion of this system
through small angles as this physical pendulum oscillates about
the pivot P.
If the maximum displacement is 2 = 0.200 rad, calculate the
total energy in the oscillations.
M = 5.20 kg; R = 4.25 cm; L = 1.25 cm
40.
A mass of 3.75 kg is attached to massless spring of force constant 1250 N/m. It slides on a horizontal
frictionless track. At t = 0 the velocity of the block is -4.25 m/s., and its position is -0.170 m. Assume a
Hooke's Law spring.
(a)
(b)
(c)
(d)
Calculate the amplitude of the motion.
Calculate the maximum speed for the motion.
W rite a complete solution in the form x = D cos ( Tt - N) with D, T and N numerically evaluated.
Be sure to include the sign of N.
W rite a complete solution in the form x = A cos Tt + B sin Tt, with A, B and T numerically
evaluated.
41.
(a)
(b)
(c)
(d)
(e)
42.
A truck with wheels 48.0 inches in diameter is traveling at 60.0 mi/hour. Calculate the angular velocity
of its wheels assuming they are not slipping
Two uniform solid spheres of mass M and radius R are touching.
Determine the moment of inertia of this system for rotation about an
axis through point A and perpendicular to the paper.
The angular frequency of an oscillator is 375.0 rad/s. Find its period.
Calculate the total kinetic energy, Joules, of a 16.1 pound bowling ball rolling without sliding and at a
speed of 40.0 ft/s.
A pendulum clock keeps good time on earth. It is taken to the moon, and started at 12:00 noon in Salt
Lake City. W hen it is 1:00 p.m. in Salt Lake City, what time does this clock read?
A cylinder of mass M and radius R is pulled by a rope wrapped around it. The force F on the
rope is parallel to the plane. The cylinder rolls up the hill without sliding. The coefficients of
friction are: :s = 2/3, :k = ½.
(a)
(b)
Calculate the linear acceleration of the cylinder up the hill. (F is large enough that it does
go up hill.)
Find the force of friction between the cylinder and the plane including direction. Take
uphill as positive.
43.
A small cylinder of radius r and mass m rolls without sliding
starting from rest at the top of a large cylinder of radius R.
Calculate the value of the angle 2 at which the cylinder is no longer
touching the large cylinder. (Numerical value. You do have
enough information.)
44.
A mass of 4.40 kg is attached to massless spring of force constant 900
N/m. It slides on a horizontal frictionless track. At t =0 the velocity of
the block is -3.25 m/s., and its position is +0.220 m. Assume a
Hooke's Law spring.
(a)
(b)
(c)
(d)
Calculate the amplitude (maximum value of x of the motion).
Calculate the maximum speed for the motion.
W rite a complete solution in the form x = D cos ( Tt - N) with D,
T and N numerically evaluated. Be sure to include the sign of
N.
W rite a complete solution in the form x = A cos Tt + B sin Tt,
with A, B and T numerically evaluated.
Note: Be sure to do those parts that you can do, even if you cannot do everything.
Data: Use these constants (where it states for example, 1 ft, the 1 is exact for significant figure purposes).
1 ft = 12 in (exact)
1 m = 3.28 ft
1 mile = 5280 ft (exact)
1 hour = 3600 sec = 60 min (exact)
1 day = 24 hr (exact)
g ea rth = 9.80 m/s 2
= 32.2 ft/s 2
g m oon = 1.67 m/s 2
= 5.48 ft/s 2
1 year = 365.25 days
1 kg = 0.0685 slug
1 N = 0.225 pound
1 horsepower = 550 ftApounds/s
(exact)