Download Unit 3 AP Universal Gravitation, Uniform Circular Motion, and

Universal Gravitation Practice Problems
rE = 6.38 E6 m
Fg=
G= 6.67 E -11 N m2/ kg2
ME= 5.98 E 24 kg
GmM
r2
* = Problems we do together in class; all others are for homework.
1. *The mass of one small ball is 1.50 g, and the mass of another is 870.0 g. If the center-to-center distance between
these two balls is 10.0 cm, find the magnitude of the gravitational force that each exerts on the other. (8.70 E-12 N)
2. On earth, two parts of a space probe weigh 11,000 N and 3,400 N. These parts are separated by a center-to-center
distance of 12m and may be treated as uniform spherical objects. Find the magnitude of the gravitational force that
each part exerts on the other out in space, far from any other objects. (1.80 E-7 N)
3. *Mars has a mass of 6.46E23 kg and a radius of 3.39E6 m. (a) What is the acceleration due to gravity on Mars? (b)
How much would a 65-kg person weigh on this planet? (3.75 m/s/s, 243.7 N)
4. Saturn has an equatorial radius of 6.00E7 m and a mass of 5.67E26 kg. (a) Compute the acceleration of gravity at the
equator of Saturn. (b) What is the ratio of a person’s weight on Saturn to that on earth? (10.51 m/s/s, 1.07x larger)
5. A neutron star has a mass of 2.0E30 kg (about the mass of our sun) and a radius of 5.0E3 m (about the height of a
good-sized mountain). Suppose an object falls from rest near the surface of such a star. How fast would it be moving
after it had fallen a distance of 0.010 m? (Assuming that the gravitational force is constant over the distance of the fall,
and that the star is not rotating.) (3.27 E5 m/s)
6. Two objects attract each other with a gravitational force of 1 E -8 N when separated by 20 cm. If the total mass of the
objects is 5 kg, what is the mass of each? (Will require the quadratic formula; 3 kg and 2 kg)
7. An astronaut orbits the earth in a space capsule whose height above the earth is equal to the twice earth's radius.
a. How does the mass of the astronaut in the capsule compare to her mass on the earth? (The same. Duh.)
b. How does the weight of the astronaut compare to her weight on earth? (g will be 1/9th the value at that
radius; therefore mg will be 1/9th the value at that radius)
Centripetal Acceleration Practice Problems
v
d
2  r
 vc 
t
T
ac 
v2
r
Fc  mac  Fc  m
v2
r
 2  r 
m

T 

=
r
2
8.
*The wheel of a car has a radius of 0.29 m and is being rotated at 830 revolutions per minute (rpm) on a tire balancing
machine. Determine
a. the speed (in m/s) at which the outer edge of the wheel is moving. (25.21 m/s)
b. the centripetal acceleration of a particle on the outer edge of the wheel and express your answer in terms of g.
(2190.84 m/s/s or ~224g)
9. *You are holding a model airplane on a guideline and swinging it around your head. If the string suddenly breaks,
what will be the new motion of the plane? (in a straight line…no more circle)
10. The tips of the blades in a food blender are moving with a speed of 21 m/s in a circle with a radius of 0.053m. How
much time does it take for the blades to make one revolution? (1.59 E-2 sec/rev)
11. The bobsled track at the 1994 Olympics in Norway contained successive turns with radii of 33m and 24 m. Find the
centripetal acceleration at each turn for a speed of 34 m/s. Express your answer as multiples of g. (35.03
m/s/s3.57g; 48.17 m/s/s 4.91g)
r = 33m
r=
24m
Unit 3 AP Practice Problems
Universal Gravitation, Centripetal Acceleration, and a little Torque
1
12. A grinding wheel (radius 7.6 cm) is rotating at 1750 rpm.
a. What is the speed of a point on the outer edge of the wheel? (13.92 m/s)
b. What is the centripetal acceleration of the point? (2.55 E3 m/s/s)
13. An airplane is flying in a horizontal circle of radius 1.5 km with a speed of 450 km/h. What is the magnitude of the
centripetal acceleration of the plane? (10.42 m/s/s)
14. *A car moves at constant speed with three parts to its motion. It moves along a straight line toward a circular turn,
goes around the turn, and then moves away along a straight line. In each of these parts is the car in equilibrium? (yes,
no, yes)
Horizontal Circular Motion and the centripetal force
15. *An airplane is flying in a horizontal circle of radius of 1.0 km.
a. What must be the speed of the plane if the pilot is to experience a centripetal acceleration of 3g? (171.46
m/s)
b. If the pilot has a mass of 75 kg, what centripetal force acts on him? (2.21 E3 N)
c. What provides the center-seeking force?
16. *A 55 kg ice skater is moving at 4 m/s when she grabs the loose end of a rope (the opposite end is tied to a pole). She
then moves in a circle of radius 0.8 m around the pole. Ignore friction.
a. Determine the force exerted by the horizontal rope on her arms. (1100N)
b. Compare this force with her weight. (~2x her weight)
17. A model airplane has mass of 0.90 Kg and moves on a 17m guideline at constant speed in a circle that is parallel to the
ground. Find the tension in the guideline for speeds of 19m/s and 38 m/s. (19.11 N, 76.45 N)
18. An athlete swings a 5 kg ball horizontally on the end of an 0.8 m long rope. The ball moves through 0.5 revolutions
per second. What is
a. The speed of the ball? (2.51 m/s)
b. The centripetal acceleration of the ball? (7.90 m/s/s)
c. If the maximum tension the rope can withstand before breaking is 100N, what is the maximum speed the ball
can have? (4 m/s)
19. *A 0.5 Kg ball that is tied to the end of a 1.5 m cord is revolved in a horizontal plane with
the cord making a 300 angle to the vertical.
a. Determine the ball’s speed. (2.06 m/s)
b. If instead the ball is revolved so that its speed is 4 m/s, what angle does the cord
make with the vertical? (XX)
c. If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at
which the ball can move? (XX)
20. *A curve in a stretch of highway has a radius R. The coefficient of static friction between
the tires and the road is us.
a. In terms of the given variables and any fundamental constants, what is the fastest speed that a car can safely
travel around the curve?
b. Explain what happens when a car enters the curve at a speed greater than the max safe speed.
21. *A certain light truck can go around a flat curve having a radius of 150 m with a max speed of 32 m/s without sliding
off the road. With what max speed can it go around a curve having a radius of 75m? (22.63 m/s)
22. *Compare the maximum speeds at which a car can safely negotiate an un-banked turn of radius 50m in dry weather
(coefficient of static friction = 0.900) and icy weather (coefficient of static friction = 0.100) without sliding off the
road. (21 m/s, 7 m/s)
23. *A stunt motorcyclist moves in a horizontal circle inside a large vertical drum which has a radius of 10.0m. If the
coefficient of static friction between the tires and the drum is 0.60, what is the smallest possible speed
that the motorcycle and rider can have and still remain in contact with the wall? (12.78 m/s)
24. The Rotar is an amusement park ride where people stand against the inside of a cylinder. Once the
cylinder is spinning fast enough the floor drops out. And then you throw up.
a. What force keep the people from falling out the bottom of the cylinder? (static friction)
b. If the coefficient of friction is 0.40 and the cylinder has a radius of 2.5 m, what is the
minimum speed of the cylinder at the wall so that the people don’t fall out? (7.83 m/s)
25. In an amusement park ride, a small child stands against the wall of a cylindrical room that is then made
to rotate. The floor drops downward and the child remains pinned against the wall. Then the child
throws up. If the radius of the device is 3.34 m and the relevant coefficient of friction between the child
and the wall is 0.47, with what minimum speed is the child moving if he is to remain pinned against the
wall? (8.35 m/s)
26. If the radius of a Rotar (vomit) ride is 3.0 m and the speed is 15 m/s, what minimum coefficient of static friction is
necessary to make sure that the people on the ride don’t fall through the bottom? (0.13)
Unit 3 AP Practice Problems
Universal Gravitation, Centripetal Acceleration, and a little Torque
2
27. 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 kg is tied
to it. The suspended mass is in equilibrium while the puck on the table
revolves.
a. What is the tension in the string? (9.8 N)
b. What is the horizontal force on the puck? (9.8 N)
c. What is the speed of the puck? (6.26 m/s)
28. A 50 kg child stands at the rim of a merry-go-round of radius 2m rotating with a speed of 6 m/s.
a. What is the child’s centripetal acceleration? (18 m/s/s)
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? (900 N)
c. What minimum coefficient of static friction is required to keep her on the ride? Is this reasonable (will she
stay on the ride)? (1.84, no it is not reasonable and she will be thrown off)
Vertical Circular Motion and the centripetal force
29. *A child swings a rock of mass m in a vertical circle using a rope of length L. The rock moves with a constant speed
v. In terms of the given variables, find the tension in the rope at the highest and lowest points.
30. *A pendulum is 0.80 m long and the bob has a mass of 1.0 kg. At the bottom of its swing, the bob’s speed is 1.6 m/s.
a. What is the tension in the string at the bottom of the swing? (13 N)
b. Explain why the tension is greater than the object’s weight at that point.
31. A 35 kg child swings on a rope with a length of 6.5 m that is hanging from a tree. At the bottom of the swing, the
child is moving at a speed of 4.20 m/s. What is the tension in the rope? (437.89 N)
32. *Picture a 600 kg motorcycle moving inside of a vertical track of radius 25 m. What is the minimum speed the
motorcycle must have in order to remain in contact with the track at the top of the path? (15.65 m/s)
33. *Picture the same motorcycle (600 kg) now moving over the crest of a hill (radius 52m). What is the maximum speed
the motorcycle can have and still remain in contact with the ground? (22.57 m/s)
34. A roller coaster has a vertical loop with a radius of 29.5 m. With what minimum speed should the roller coaster car be
moving inside the loop at the top of the track so that the passengers don’t lose contact with the seats? (17 m/s)
35. *A motorcycle has a constant speed of 25.0 m/s as it passes over the top of a hill whose radius of curvature is 126m.
The mass of the motorcycle and rider is 342 kg. Find the magnitude of
a. The centripetal force (remember, this is a net force). (1.7 E3 N)
b. The normal force that acts on the motorcycle. (1.66 E3 N ---vs. 3.35 E3 N for its weight)
36. A car approaches the top of a hill that is shaped like a vertical circle with a radius of 55m. What is the fastest speed
that the car can go over the hill without losing contact with the ground? (23.22 m/s)
37. A 1.5 kg toy motorcycle is moving on the inside of a vertical circular track (r = 1.0m). It arrives at the top of the
track with a speed of 5.0 m/s.
a. What force does the track exert on the motorcycle at the top of the track? (22.8 N)
b. What force does the track exert on the motorcycle when it is at the bottom of the track? (52.2 N)
Satellites and Artificial Gravity
rE = 6.38 E6 m
in orbit:


G= 6.67 E -11 N m2/ kg2
GmM
“ Fnet = Fc = mg” =
r2
= m  ac =
ME= 5.98 E 24 kg
mv
r
2
=
 2  r 
m

 T 
r
2
All satellites are in free-fall where the only force acting on them is their weight, mg (but g will be different at the
height the satellite is orbiting).
In order to produce a sense of artificial gravity, you would need to create an a c that is equal to g.
38. *Determine the speed of the Hubble space telescope orbiting at a height of 598 km above the earth’s surface. (7.56
E3 m/s)
39. *The Hubble telescope has detected the light being emitted from different regions of galaxy M87. From the
characteristics of this light, astronomers have determined an orbiting speed of 7.5 E 5 m/s for matter located a distance
of 5.7 E 17 m from the center. Find the mass of the object located at the galactic center. (4.81 E39 kg)
40. *What is the height above the earth’s surface at which all geosynchronous satellites (a satellite that orbits directly
above a single position on the earth so its period is the same as the earth’s 24 hours), regardless of mass, must be
placed in orbit? (3.59 E7 m)
Unit 3 AP Practice Problems
Universal Gravitation, Centripetal Acceleration, and a little Torque
3
41. *A spacecraft orbits Mars (mass= 6.40 E23 kg) in a circular orbit of radius 8.01 E5 km. What is the period of the
spacecraft? (2.18 E7 sec or 252 days  that’s kind of big so it must be super far from the surface of the planet)
42. A satellite moves in a circular orbit around the Earth at a speed of 5,000 m/s. Determine
a. The satellite’s altitude above the surface of Earth. (9.58 E6 m)
b. The period of the satellite’s orbit. (5.57 hour)
43. A satellite is placed in a circular orbit. What must be the height above the surface of the earth of its orbit if it is to
circle the earth one complete pass each day? (3.59 E7 m)
44. The moon of a planet is observed to have a nearly circular orbit, r= 4 E5 km, and an orbital period of 21.5 days. What
is the mass of the planet? (1.09 E25 kg)
45. *At what speed must the surface of a space station (r = 1700 m) move so that an astronaut on the interior outer-surface
experiences a push on his feet that equals his earth weight? (129.07 m/s)
46. *How many revolutions per minute must a rotating space station (r = 1200 m) turn to provide an artificial gravity of
0.50g? (0.61 rev/min)
47. A space station is shaped like a ring and rotates to simulate gravity. If the radius of the space station is 120 m, what
period must it have so that is simulates Earth’s gravity? (21.99 sec/rev)
48. If a washing machine’s drum has a radius of 25 cm and spins at 4 rev/sec, what is the strength of the artificial gravity
which the clothes are subjected? Express your answers in terms of g. (157.91 m/s/s or ~16g)
Torque Practice Problems
  F 
49. *In order for an object to be in a state of equilibrium, what must be true of
a. the forces/torques acting on it? (they must all balance so there is no NET force acting on the object)
b. Its velocity? (constant)
c. Its acceleration? (zero)
50. *When a net torque is applied to an object in equilibrium, what happens to the object’s
a. Equilibrium? (goes away---no longer in equilibrium)
b. Acceleration? (an acceleration must now be present)
c. linear velocity? (zero--- the net torque will not cause the object to move in a straight line; the object
will instead rotate about an axis)
d. angular velocity? (it gets what is called an “angular velocity” but we don’t study that in AP B)
51. *A wrench is used to tighten a nut. If a 14 N force is applied perpendicularly 15.0 cm from the axis of rotation, what
is the torque due to the applied force? (2.1 N m)
52. *A 20 N force is applied to a door 0.52 meter from the hinge (left side of drawing) as shown in the drawing below.
The force is applied at an angle of 620 to the door. At the instant shown, what is the torque on the door? (9.18 N m)
53. A mass-less meter stick is pivoted at the 0.50-m line. A 5.0-kg object is hung from the 0.15 m line. Where should a
13.0 kg object be hung to achieve equilibrium? (at the 0.63 m mark)
54. A 55 kg kid and a 75 kg kid are placed at opposite ends of a massless beam supported by a fulcrum at some point in
between the two so that the system is in equilibrium. What must be the ratio of the lengths between each child and the
fulcrum? (the 75 kg kid is only 0.73x the distance to the fulcrum than the 55 kg kid)
55. A 76 kg man balances a boy on a massless see-saw. The boy is 3.2 m away from the fulcrum and the man is 1.43 m
away from the fulcrum.
a. What is the mass of the boy? (33.96 kg)
b. What is the magnitude of the upward force the fulcrum exerts on the see-saw? (1.08 E3 N)
Unit 3 AP Practice Problems
Universal Gravitation, Centripetal Acceleration, and a little Torque
4