REVIEW: LUG & UCM

Planet X has a mass of M and a radius of R. Planet Y has a mass of 3M and a radius of 3R. Identical satellites orbit both planets at a distance R above their surfaces, as shown. The planets are separated by such a large distance that the gravitational forces between them are negligible.

How does the magnitude of the gravitational force FY exerted by Planet Y on its satellite compare to the gravitational force FX exerted by Planet X on its satellite?

• FY=3FX

Answer explanation

Planet X has a mass of M and a radius of R. Planet Y has a mass of 3M and a radius of 3R. Identical satellites orbit both planets at a distance R above their surfaces, as shown above. The planets are separated by such a large distance that the gravitational forces between them are negligible.

How does the gravitational field g_X at the surface of Planet X compare with the gravitational field g_Y at the surface of Planet Y?

• g_X=9g_Y

• g_X=3g_Y

Answer explanation

A moon orbits a planet in a nearly circular orbit of radius R, as shown.

Which of the following free-body diagrams could be used to analyze the forces exerted on the moon when it is at the position indicated in the figure?

• 
• 
• 
• 

Answer explanation

The only force exerted on the moon is the force due to gravity from the planet. This force is directed toward the center of the planet, or toward the right.

A moon orbits a planet in a nearly circular orbit of radius RR, as shown.

The moon has a mass of 1 ×10²² kg, and the gravitational field strength at a distance R from the planet is 0.001 N/kg. What is the gravitational force exerted on the moon while it is in orbit around the planet?

• 0 N

• 1 ×10¹⁹ N

• 1 ×10²² N

• 1 ×10²⁵N

Answer explanation

To determine the force due to gravity exerted on the moon, the planet’s mass must be multiplied by the gravitational field strength of the planet at the moon’s location.

A moon orbits a planet in a nearly circular orbit of radius RR, as shown.

Astronomers making careful observations of the moon’s orbit discover that the orbit is not perfectly circular, nor is it elliptical. Which of the following statements supports this observation?

• The moon and the planet exert forces of equal magnitude on each other.

• There is another celestial body that exerts a gravitational force on the moon.

• The value of the gravitational constant G is different in the location near the planet-moon system.

• There is a centripetal force that causes the net force exerted on the moon to be different from the gravitational force.

Answer explanation

If the only force exerted on the moon was from the planet, it would be expected that the planet would travel in a circular or elliptical orbit. However, since the path is neither circular nor elliptical, the mass from another celestial body must exert a force on the moon in addition to the force exerted by the planet.

Two containers of water can have their individual masses varied by adding or removing water. The containers are initially a distance d apart, as measured from their centers, and are filled with water so that each has a mass M, as shown in Figure 1. The gravitational force that one container exerts on the other is F₀. Water is then added to one container so that its mass increases to 1.5M, and water is removed from the other container so that its mass decreases to 0.5M, as shown in Figure 2. What is the new gravitational force exerted on one container by the other?

• F₀

• 2F₀

Answer explanation

Satellite A orbits a planet at a distance d from the planet’s center with a centripetal acceleration a₀. A second identical satellite B orbits the same planet at a distance 2d from the planet’s center with centripetal acceleration a_b. What is the centripetal acceleration a_b in terms of a₀ ?

• 2a₀

• 4a₀

Answer explanation

A student must conduct two experiments so that the inertial mass and gravitational mass of the same object can be determined. In the experiment to find the object's gravitational mass, the student ties one end of a string around the object with the other end tied to a spring scale so that the object can vertically hang at rest. In the experiment to find the object's inertial mass, the student uses a spring scale to pull the object, starting from rest, across a horizontal surface with a constant applied force such that frictional forces are considered to be negligible. In addition to the spring scale, the student has access to other measuring devices commonly found in a science laboratory. Which of the following lists the essential measuring devices the student can use to collect the data necessary to find the object's gravitational and inertial mass?

• Meterstick and timer

• Meterstick, timer, and motion detector

• Meterstick, timer, motion detector, and mass balance

• Meterstick, timer, motion detector, mass balance, and protractor

Answer explanation

Only the spring scale is required to find the gravitational mass of the object as it hangs from the spring scale. In the experiment to find the object's inertial mass, the meterstick can be used to measure the total displacement of the object as it is pulled with the spring scale. The timer can be used to determine the total time in which the object is displaced from its initial position to its final position. With information pertaining to the object's total displacement and time of travel, the appropriate kinematics equation can be used to determine the acceleration of the object. Once the acceleration of the object is known, an application of the equation of Newton's second law of motion can be used to determine the inertial mass of the object.

Two experiments are conducted are conducted to determine the mass of an object. In Experiment 1, the object's weight is measured by using an electronic balance once the object has been placed at rest on the balance. In Experiment 2, the object is pulled along a horizontal surface with a spring scale such that the force reading on the spring scale remains constant while a motion detector is used to measure the instantaneous speed of the object as it is pulled. All frictional forces for both experiments are considered to be negligible. Which of the two experiments, if either, could be used to determine the gravitational mass of the object?

• Experiment 1 only

• Experiment 2 only

• both experiments

• neither experiments

Answer explanation

The gravitational mass of an object can be determined in an experiment in which a mass has been placed in a gravitational field or the net force on a mass is the result of a gravitational force. Therefore, the object's weight is the result of the gravitational force exerted on the object.

A student analyzes data of the motion of a planet as it orbits a star that is in deep space. The orbit of the planet is considered to be stable and does not change over time. The student claims, “The only experimentally measurable external force exerted on the planet is the force due to gravity from the star.” Is the student’s claim supported by the evidence? What reasoning either supports or contradicts the student’s claim?

• Yes. Other external forces are exerted on the planet, but they are of negligible magnitude.

• Yes. The only external force exerted on the planet is the force due to gravity from the star.

• No. The strong nuclear forces, weak nuclear forces, and the electromagnetic forces exerted on the planet could be measured.

• No. There are no external forces exerted on the planet.

Answer explanation

The evidence provided supports the claim that there must be a gravitational force exerted on the planet from the star for the planet to orbit the star in a stable orbit. While other interstellar objects may exert gravitational forces on the planet, for a typical, stable orbit, they are of negligible magnitude.