Deck 7: Circular Motion and Gravitation

Question 1

A planet revolves clockwise around a star, with constant speed
Figure 7-3

Which graph shows the direction of the planet's acceleration at point P:
graph a, graph b, graph c, or graph d?

Answer 1

graph d

Question 2

A girl attaches a rock to a string, which she then swings counter-clockwise in a horizontal circle. The string breaks at point P in Figure 7-4, which shows a bird's-eye view (i.e., as seen from above) Figure 7-4

Which path (A-E) will the rock follow?

Answer 2

path B

Question 3

Imagine yourself in a car traveling around a sharp curve. The "centripetal" force is inward, yet why do you slide outward and not inward?

Answer 3

In the absence of an inward force, you want to continue in a straight line which would cause you to move "outward". The frictional force of the seat or the push of the side of the car pushes you "inward" and supplies the centripetal force to keep you on your circular path.

Question 4

The weight of an object on the Moon is what fraction of the weight of the same object on Earth?

Question 5

Show that the change in gravitational potential energy U₂ -U₁ = -GmM(1/R₂ - 1/R₁) becomes mgh when h << r.

Question 6

Why do the two different expressions for gravitational potential energy (+mgh and -GmM/r) have Different signs?

Question 7

State Kepler's first law.

Question 8

State Kepler's second law.

Question 9

State Kepler's third law.

Question 10

How was Newton able to calculate the period of revolution of the Moon using his gravitation theory, even though he didn't know the value of G or the masses of the Earth and Moon?

Question 11

To test his gravitational theory, Newton calculated the period of the moon around the Earth to see if he got the observed 27.3 days. Section 7.6 derived T = 2π . How could Newton calculate T when he didn't know the value of G or of M?

Question 12

Newton never knew the numerical value of "his" universal gravitational constant G.

Question 13

The net force on an object in orbit is zero.

Question 14

The constant K in Kepler's Third Law is the same for all the planets in our solar system, but is different for planetary systems orbiting other stars.

Question 15

Marilyn (M) Figure 7-1

Which of the following is true?

A) They have different speeds, and different angular velocities.
B) They have the same speed and the same angular velocity.
C) They have the same speed, but their angular velocity is different.
D) They have different speeds, but the same angular velocity.

Question 16

James is twirling a rock around a horizontal circle on the end of a string. Figure 7-2 is a top view (motion is counterclockwise) Figure 7-2

If the speed is constant what is the acceleration at point A?

A) zero
B) centrifugal
C) r

D) v / r
E) centripetal

Question 17

A pilot executes a vertical dive. Just before the plane starts to come up from the bottom of the dive, the force on him is

A) more than g, and pointing up.
B) less than g, and pointing down.
C) equal to g, and pointing down.
D) more than g, and pointing down.
E) less than g, and pointing up.

Question 18

A car, driven around a circle with constant speed, must have

A) a tangential acceleration.
B) zero acceleration.
C) zero velocity.
D) a centripetal acceleration.
E) a centrifugal acceleration.

Question 19

The speed needed at the bottom of a loop-the-loop track so that a car can coast to the top, with sufficient speed to stay on the track, depends on the mass of the car. (Neglect the effects of friction.)

A) sometimes true, since kinetic energy is a function of mass
B) never true
C) sometimes true, since potential energy is a function of mass
D) always true, since kinetic energy is a function of mass
E) always true, since potential energy is a function of mass

Question 20

Does the centripetal force acting on an object do work on the object?

A) yes, because the force and the displacement of the object are perpendicular
B) yes, since it takes energy to turn an object
C) yes, since a force acts and the object moves, and work is force times distance
D) no, because the force and the displacement of the object are perpendicular
E) no, because the object has constant speed

Question 21

Two horizontal curves on a bobsled run are banked at the same angle, but one has twice the radius of the other. The safe speed (no friction needed to stay on the run) for the smaller radius curve is V. Therefore, the safe speed on the larger radius curve is

A) 2. V.
B) 0.71 V.
C) 0.5 V.
D) 1.4 V.
E) 0.10 V.

Question 22

The record playing on the turntable of your stereo is rotating clockwise (as seen from above in Figure 7-5) Figure 7-5


The direction of the acceleration of point P (at the left) is

A) graph a.
B) graph b.
C) graph c.
D) graph d.
E) graph e.

Question 23

A race car, traveling at a constant speed of 50 m/s, drives around a circular track of radius 250 m. What angular acceleration does it experience?

A) 1.0 × 10¹ rad/s²
B) zero
C) 1.0 × 1⁰⁴ rad/s²
D) 6.3 × 1⁰² rad/s²
E) 2.0 × 1⁰³ rad/s²

Question 24

A moon is in orbit around its mother planet. Which of the following statements is always true about its kinetic energy (K), and its gravitational potential energy (U)?

A) K is positive, and U is positive.
B) K is negative, and U is negative.
C) K is positive, and U is negative.
D) K is negative, and U is positive.

Question 25

For a spherical planet to act like a "point mass", so that externally F = GmM/r², it

A) must have spherical mass symmetry.
B) must have a mass << r c² / (2G).
C) can not be hollow inside.
D) must have constant density throughout.

Question 26

A spherically symmetric planet has four times the Earth's mass and twice its radius. If a jar of peanut butter weighs 12. N on the surface of the Earth, how much would it weigh on the surface of this planet?

A) 30 N
B) 24 N
C) 12 N
D) 6 N
E) 18 N

Question 27

Two asteroids, of the same mass, have a gravitational potential energy U. If a third asteroid, of the same mass, moves toward them, to a position such that the distances between the three asteroids are the same, the gravitational potential energy of the system will have changed by

A) 4U.
B) U.
C) 2U.
D) 3U.
E) 5U.

Question 28

Who was the first person to realize that the planets move in elliptical paths around the sun?

A) Galileo
B) Copernicus
C) Kepler
D) Einstein
E) Brahe

Question 29

The speed of Halley's comet, while traveling in its elliptical orbit around the sun,

A) increases as it nears the sun.
B) is constant.
C) is zero at two points in the orbit.
D) is zero at four points in the orbit.
E) decreases as it nears the sun.

Question 30

Kepler's discovery that T²/r³ = K applies

A) to orbits where K is a universal constant.
B) only to circular orbits.
C) to elliptical orbits where r is the average distance.
D) to lunar motion provided K is the same for planetary motion.

Question 31

The speed for a "low" circular orbit about the Earth is about

A) 7.6 km/s.
B) 2.5 × 10⁴ km/s.
C) 2.1 km/s.
D) 1.7 × 10⁴ km/s.
E) 17. km/s.

Question 32

If the Earth had four times its present mass, what would be its new period of revolution around the sun, compared to its present orbital period?

A) four times as much
B) two times as much
C) the same
D) one-half as much
E) one-fourth as much

Question 33

A planet is discovered to orbit around a star in the galaxy Andromeda, with the same orbital diameter as the Earth around our sun. If that star has 4 times the mass of our sun, what will the period of revolution of that new planet be, compared to the Earth's orbital period?

A) four times as much
B) times as much
C) one-fourth as much
D) twice as much
E) one-half as much

Question 34

The following statements refer to man-made, artificial satellites in orbit around Earth. Which is an accurate statement?

A) The velocity required to keep a satellite in a given orbit depends on the mass of the satellite.
B) Only circular orbits are possible for artificial satellites.
C) The period of revolution of a satellite moving about the Earth is independent of the size of the orbit it travels.
D) A satellite in a large diameter circular orbit will always have a longer period of revolution about the Earth than will a satellite in a smaller circular orbit.
E) It is possible to have a satellite traveling at either a high speed or at a low speed in a given circular orbit.

Question 35

Kepler's 3rd law says T²/r³ = K. What does the constant K depend upon?

A) size
B) temperature
C) mass
D) frequency
E) period

Question 36

Consider a small satellite moving in a circular orbit (radius r) about a spherical planet (mass M). The period does not depend upon

A) the radius r.
B) the planet mass.
C) g at the satellite position.
D) the universal gravitational constant.
E) the satellite mass.

Question 37

The total energy of a circularly orbiting satellite is ________ the kinetic energy.

A) three times
B) twice
C) the same as
D) half
E) times

Question 38

Satellite A has twice the mass of satellite T, and rotates in the same orbit. Which of the following is true?

A) The speed of T is half the speed of A.
B) The speed of T is three-fourths the speed of A.
C) The speed of T is one-fourth the speed of A.
D) The speed of T is equal to the speed of A.
E) The speed of T is twice the speed of A.

Question 39

Consider an Earth satellite in an elliptical orbit. Its centripetal acceleration is

A) constant.
B) zero.
C) < g at the satellite.
D) g at the satellite.
E) > g at the satellite.

Question 40

Your starship, the TOP-QUARK, is in a counter-clockwise "parking orbit" around the planet VULCAN, as shown in Figure 7-6.
Figure 7-6

In order to achieve escape velocity, the direction your retro-rockets should be facing is

A) graph a
B) graph b
C) graph c
D) graph d
E) none of the above.

Question 41

The escape velocity is how many times the orbital velocity?

A) 4
B) 2

C) 2
D)

E)

Question 42

What is the arc length subtended by an angle of 30° on a circle of radius 10 cm?

Question 43

A triangle has angles of 1.10 radians and 0.25 radians. What is the third angle, expressed in radians and in degrees?

Question 44

A bicycle wheel has an outside diameter of 66 cm. How far does a point on the rim travel when the wheel rotates through an angle of 70°?

Question 45

What is the angular speed in rad/s of an electric motor that rotates at 1.80 × 10³ rpm?

Question 46

What is the angular speed of a point on the earth's surface at 60.0° north latitude? What is the linear speed of such a point?

Question 47

A pilot makes an outside vertical loop of radius 320 m. At the top of his loop he is pushing down on his seat with only one-half of his normal weight. How fast is he moving?

Question 48

The maximum acceleration a pilot can stand is about 7.0 g. What is the minimum radius of curvature that a jet plane's pilot, pulling out of a vertical dive, can tolerate at a speed of 250 m/s?

Question 49

Jenny drives at a constant speed of 15. m/s around a circular drive of diameter 60. meters.
(a) Her acceleration is ________ m/s2.
(b) Is her acceleration tangential, centripetal, or centrifugal?
(c) Is her velocity tangential, centripetal, or centrifugal?

Question 50

A coin of mass m rests on a turntable a distance r from the axis of rotation. The turntable rotates with frequency f. Derive an expression for the minimum coefficient of friction between the turntable and the coin if the coin is not to slip.

Question 51

Fred swings around a 0.40 kg ball attached to a string at constant speed in an almost horizontal circle of radius 1.3 meters. If one revolution takes 1.4 s:
(a) What is the angular speed of the ball?
(b) What centripetal force is exerted on the ball?

Question 52

A tooth polisher on a dentist's drill reaches a speed of 1800 rpm in 8.0 s. What is its average angular acceleration?

Question 53

Mary is 3.00 meters from the center of a merry-go-round. Her tangential speed is 1.88 m/s.
(a) What is her angular speed in rad/s?
(b) What is her acceleration while her speed remains 1.88 m/s?

Question 54

A turntable was revolving at 33 1/3 rpm. It was shut off and uniformly slowed down and stopped in 5.5 seconds.
(a) What was the angular acceleration?
(b) Through how many revolutions did it turn?

Question 55

Consider a merry-go-round which makes 20. revolutions in 4.0 minutes.
(a) What is the average angular speed in radians/s?
(b) For a child 3.5 m from the center, how far does he travel in the 4.0 minutes?

Question 56

Consider the planet Mars:
[Earth mass = 6.0 × 10²⁴ kg = 9.3 × Mars mass;
Earth radius = 6380.km = 1.89 × Mars radius]
What is the acceleration of gravity on Mars?

Question 57

By how many Newtons does the weight of a 100. kg person change when he goes from sea level to an altitude of 5.00 km?

Question 58

What is the gravitational force on a 70. kg person on the Earth, due to the moon? The mass of the moon is 7.36 x 10²² kg and the distance to the moon is 3.82 × 10⁸ m.

Question 59

For a spacecraft going from the earth toward the sun, at what point will the gravitational forces due to the sun and the Earth cancel?
[Earth's mass: me = 5.98 × 10²⁴ kg
sun's mass: ms = 1.99 × 10³⁰ kg
Earth-sun distance: r = 1.50 × 10¹¹ m]

Question 60

Consider a 2.4 kilogram feather (Big Bird) on the planet Mars. [Earth mass = 6.0 × 10²⁴ kg = 9.3 × Mars mass; Earth radius = 6380. km = 1.89 × Mars radius]
(a) What does the feather weigh on Mars?
(b) What would the same feather weigh on Earth?

Question 61

A 63. kg astronaut walks upon the surface of the planet Krypton which has 100. times the mass of the Earth and 100. times the diameter of the Earth. What does she weigh on Krypton?

Question 62

An astronaut lands on a previously unknown planet of radius 2350. km and she measures the acceleration of gravity to be 3.20 m/s².
(a) What is the mass of this new planet?
(b) What does the astronaut weigh on this new planet if she weighs 384. N on Earth?

Question 63

A mass which weighs 19.6 Newtons on Earth is 2.3 meters away from a 2.00 kg mass.
(a) Determine their mutual gravitational attraction to each other.
(b) How much work would have to be done to put a very large distance between the two masses?

Question 64

Our second closest star, Alpha Centauri, is 4.0 lightyears distant. On a particular night, it is located 37.° up from the horizon, in a vertical plane. What are its x and y coordinates in this plane?

Question 65

The diameter of the sun is 13.9 × 10⁸ m, and the distance from the Earth to the sun is 1.5 × 10¹¹ m. How long does it take for the sun to disappear after it touches the horizon?

Question 66

As you know, the Earth has an orbital period of one year at an orbital radius of one AU. If a new "minor planet" were to be found in a circular orbit with radius 13. AU, what would be its period?

Question 67

A satellite orbits the Earth once every 6.0 hours in a circle. What is its angular velocity (in radians/s)?

Question 68

An Earth satellite is in circular orbit 230 km above the surface of the Earth. It is observed to have a period of 89 min. From this information, estimate the mass of the Earth.

Question 69

Europa, a moon of Jupiter, has an orbital diameter of 1.34 × 10⁹ m, and a period of 3.55 days. What is the mass of Jupiter?

Question 70

A satellite orbits the Earth once every 6.0 hours in a circle. What is the radius of the orbit if it is a 2000. Kg satellite?

Question 71

A satellite is in a low circular orbit about the Earth (i.e., it just skims the surface of the Earth). How long does it take to make one revolution around the Earth?

Question 72

What is the escape velocity from the solar system (SUN) leaving from our position 93. million miles from the sun?

Question 73

A satellite orbits the Earth once every 6.0 hours in a circle. What is the acceleration of the satellite (direction too)?

Question 74

James moves in a circular path (radius 15.0 m) with a speed of 60.0 m/s. His ANGULAR VELOCITY is

A) 120 radians/s.
B) 120/ $\pi$ radians/s.
C) 120 $\pi$ radians/s.
D) 120/ $\pi$ rpm.
E) 120. rev/s.

Question 75

A fan blade, whose diameter is 1.0 m, is turning with an angular velocity of 2.0 rad/s. What is the tangential velocity of a point on the tip of the blade?

A) 1.5 m/s
B) 2.0 m/s
C) 5.0 cm/s
D) 0.50 m/s
E) 1.0 m/s

Question 76

A pulsar (a rotating neutron star) emits pulses at a frequency of 0.40 kHz. The period of its rotation is

A) 0.25 s.
B) 2.5 s.
C) 25 ms.
D) 2.5 ms.
E) 0.025 s.

Question 77

What is the angular speed of a record that rotates at 33-1/3 rpm?

A) 33.3 $\pi$ rad/s
B) 2.77 rad/s
C) 33.3/ $\pi$ rad/s
D) 3.49 rad/s
E) 1.74 rad/s

Question 78

A race car, traveling at a constant speed of 50. m/s, drives around a circular track of radius 250. m. What magnitude acceleration does it experience?

A) zero
B) 10. m/s²
C) 1.0 m/s²
D) 2.2 m/s²
E) 0.63 km/s²

Question 79

A flywheel, of radius 1 m, is spinning with a constant angular velocity of 2 rad/s. What is the centripetal acceleration of a point on the wheel's rim?

A) 2 m/s²
B) 8 m/s²
C) 4 m/s²
D) 6 m/s²
E) 1 m/s²

Question 80

What is the centripetal acceleration of a point on the perimeter of a bicycle wheel of diameter 70. cm when the bike is moving 8.0 m/s?

A) 91 m/s²
B) 1.8 × 10² m/s²
C) 2.1 × 10^-2 m/s²
D) 66 m/s²
E) 2.7 × 10² m/s²