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Section 7.4
Centripetal Acceleration
14.
It has been suggested that rotating cylinders about 10 mi long and 5.0 mi in
diameter be placed in space and used as colonies. What angular speed must such a
cylinder have so that the centripetal acceleration at its surface equals the free-fall
acceleration on Earth?
15.
Find the centripetal accelerations of (a) a point on the equator of Earth and (b) the
North Pole, due to the rotation of Earth about its axis.
16.
A tire 2.00 ft in diameter is placed on a balancing machine, where it is spun so
that its tread is moving at a constant speed of 60.0 mi/h. A small stone is stuck in the
tread of the tire. What is the acceleration of the stone as the tire is being balanced?
17.
(a) What is the tangential acceleration of a bug on the rim of a 10-in.-diameter
disk if the disk moves from rest to an angular speed of 78 rev/min in 3.0 s? (b) When the
disk is at its final speed, what is the tangential velocity of the bug? (c) One second after
the bug starts from rest, what are its tangential acceleration, centripetal acceleration,
and total acceleration?
18.
A race car starts from rest on a circular track of radius 400 m. The car’s speed
increases at the constant rate of 0.500 m/s2. At the point where the magnitudes of the
centripetal and tangential accelerations are equal, determine (a) the speed of the race
car, (b) the distance traveled, and (c) the elapsed time.
19.
A 55.0-kg ice-skater is moving at 4.00 m/s when she grabs the loose end of a rope,
the opposite end of which is tied to a pole. She then moves in a circle of radius 0.800 m
around the pole. (a) Determine the force exerted by the horizontal rope on her arms. (b)
Compare this force with her weight.
20.
A sample of blood is placed in a centrifuge of radius 15.0 cm. The mass of a red
blood cell is 3.0 × 10–16 kg, and the magnitude of the force acting on it as it settles out of
the plasma is 4.0 × 10–11 N. At how many revolutions per second should the centrifuge
be operated?
21.
A certain light truck can go around a flat curve having a radius of 150 m with a
maximum speed of 32.0 m/s. With what maximum speed can it go around a curve
having a radius of 75.0 m?
22.
The cornering performance of an automobile is evaluated on a skid pad, where the
maximum speed that a car can maintain around a circular path on a dry, flat surface is
measured. Then the centripetal acceleration, also called the lateral acceleration, is
calculated as a multiple of the free-fall acceleration g. The main factors affecting the
performance of the car are its tire characteristics and suspension system. A Dodge Viper
GTS can negotiate a skid pad of radius 61.0 m at 86.5 km/h. Calculate its maximum
lateral acceleration.
23. A 50.0-kg child stands at the rim of a merry-go-round of radius 2.00 m, rotating with
an angular speed of 3.00 rad/s. (a) What is the child’s centripetal acceleration? (b) What
is the minimum force between her feet and the floor of the carousel that is required to
keep her in the circular path? (c) What minimum coefficient of static friction is required?
Is the answer you found reasonable? In other words, is she likely to stay on the merrygo-round?
25.
An air puck of mass 0.25 kg is tied to a string and allowed to revolve in a circle of
radius 1.0 m on a frictionless horizontal table. The other end of the string passes
through a hole in the center of the table, and a mass of 1.0 kg is tied to it (Fig. P7.25).
The suspended mass remains in equilibrium while the puck on the tabletop revolves.
(a) What is the tension in the string? (b) What is the horizontal force acting on the puck?
(c) What is the speed of the puck?
Figure P7.25
26.
Tarzan (m = 85 kg) tries to cross a river by swinging from a 10-m-long vine. His
speed at the bottom of the swing (as he just clears the water) is 8.0 m/s. Tarzan doesn’t
know that the vine has a breaking strength of 1 000 N. Does he make it safely across the
river? Justify your answer.
27.
A 40.0-kg child takes a ride on a Ferris wheel that rotates four times each minute
and has a diameter of 18.0 m. (a) What is the centripetal acceleration of the child? (b)
What force (magnitude and direction) does the seat exert on the child at the lowest
point of the ride? (c) What force does the seat exert on the child at the highest point of
the ride? (d) What force does the seat exert on the child when the child is halfway
between the top and bottom?
28.
A roller-coaster vehicle has a mass of 500 kg when fully loaded with passengers
(Fig. P7.28). (a) If the vehicle has a speed of 20.0 m/s at point , what is the force of the
track on the vehicle at this point? (b) What is the maximum speed the vehicle can have
at point
in order for gravity to hold it on the track?
Figure P7.28
Section 7.5
Newtonian Gravitation
29.
The average distance separating Earth and the Moon is 384 000 km. Use the data
in Table 7.3 to find the net gravitational force exerted by Earth and the Moon on a 3.00 ×
104-kg spaceship located halfway between them.
30.
During a solar eclipse, the Moon, Earth, and Sun all lie on the same line, with the
Moon between Earth and the Sun. (a) What force is exerted by the Sun on the Moon? (b)
What force is exerted by Earth on the Moon? (c) What force is exerted by the Sun on
Earth? (See Table 7.3 and Problem 29.)
31.
A coordinate system (in meters) is constructed on the surface of a pool table, and
three objects are placed on the table as follows: a 2.0-kg object at the origin of the
coordinate system, a 3.0-kg object at (0, 2.0), and a 4.0-kg object at (4.0, 0). Find the
resultant gravitational force exerted by the other two objects on the object at the origin.
32.
Use the data of Table 7.3 to find the point between Earth and the Sun at which an
object can be placed so that the net gravitational force exerted by Earth and the Sun on
that object is zero.
33.
Objects with masses of 200 kg and 500 kg are separated by 0.400 m. (a) Find the
net gravitational force exerted by these objects on a 50.0-kg object placed midway
between them. (b) At what position (other than infinitely remote ones) can the 50.0-kg
object be placed so as to experience a net force of zero?
34.
Two objects attract each other with a gravitational force of magnitude 1.00 × 10–8
N when separated by 20.0 cm. If the total mass of the objects is 5.00 kg, what is the mass
of each?
Section 7.6
Kepler’s Laws
35.
A satellite moves in a circular orbit around Earth at a speed of 5
000 m/s. Determine (a) the satellite’s altitude above the surface of Earth and (b) the
period of the satellite’s orbit.
36.
A 600-kg satellite is in a circular orbit about Earth at a height above Earth equal
to Earth’s mean radius. Find (a) the satellite’s orbital speed, (b) the period of its
revolution, and (c) the gravitational force acting on it.
37.
Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of
4.22 × 105 km. From these data, determine the mass of Jupiter.
38.
A satellite has a mass of 100 kg and is located at 2.00 × 106 m above the surface of
Earth. (a) What is the potential energy associated with the satellite at this location? (b)
What is the magnitude of the gravitational force on the satellite?
39.
A satellite of mass 200 kg is launched from a site on Earth’s equator into an orbit
200 km above the surface of Earth. (a) Assuming a circular orbit, what is the orbital
period of this satellite? (b) What is the satellite’s speed in it’s orbit? (c) What is the
minimum energy necessary to place the satellite in orbit, assuming no air friction?