Book cover for University Physics with Modern Physics

University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

ISBN #9780321501219

12th Edition

3,769 Questions

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619,728 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter on gravitation provides a comprehensive look at the forces that govern the motion of objects, from everyday terrestrial phenomena to complex cosmic systems. Key principles such as Newton's Law of Gravitation and Kepler's Laws provide the foundational tools for understanding gravitational interactions, satellite orbits, and the behavior of objects in extreme conditions, such as near black holes. Grasping these concepts facilitates both theoretical and practical applications in physics and astronomy.

Learning Objectives

1

Explain the fundamental principles of gravitation, including Newton's Law of Gravitation and Kepler's Laws.

2

Analyze the concepts of gravitational potential energy and apparent weight in both terrestrial and astronomical contexts.

3

Apply the principles of gravitation to problems involving satellite motion and planetary orbits.

4

Explore advanced topics such as the gravitational dynamics of black holes.

Key Concepts

CONCEPT

DEFINITION

Gravitation

A fundamental natural force that causes objects with mass to attract each other.

Newton's Law of Gravitation

A law stating that every mass exerts an attractive force on every other mass, with the force proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Kepler's Laws

Three scientific laws describing the motion of planets around the sun, including the shape of orbits and the relation between orbital period and distance.

Gravitational Potential Energy

The energy possessed by an object due to its position in a gravitational field; energy is higher the greater the distance from the source of the gravitational field.

Apparent Weight

The weight of an object as measured under the influence of additional forces, such as acceleration or centrifugal force, differing from the true gravitational weight.

Black Holes

Extremely dense regions in space where gravitational forces are so strong that nothing, not even light, can escape from them.

Example Problems

Example 1

What is the ratio of the gravitational pull of the sun on the moon to that of the earth on the moon? (Assume the distance of the moon from the sun can be approximated by the distance of the earth from the sun.) Use the data in Appendix $F .$ Is it more accurate to say the moon orbits the earth, or that the moon orbits the sun?

Example 2

Cavendish Experiment. In the Cavendish balance apparatus shown in Fig. 12.4 , suppose that $m_{1}=1.10 \mathrm{kg}, m_{2}=25.0 \mathrm{kg}$ . and the rod connecting the $m_{1}$ pairs is 30.0 $\mathrm{cm}$ long. If, in each pair, $m_{1}$ and $m_{2}$ are 12.0 $\mathrm{cm}$ apart center-to center, find (a) the net force and (b) the net torque (about the rotation axis) on the rotating part of the apparatus. (c) Does it seem that the torque in part (b) would be enough to easily rotate the rod? Suggest some ways to improve the sensitivity of this experiment.

Example 3

How far from a very small $100-k g$ ball would a particle have to be placed so that the ball pulled on the particle just as hard as the earth does? Is it reasonable that you could actually set up this as an experiment?

Example 4

Two uniform spheres, each with mass $M$ and radius $R,$ touch each other. What is the magnitude of their gravitational force of attraction?

Example 5

An interplanetary spaceship passes through the point in space where the gravitational forces from the sun and the earth on the ship exactly cancel. (a) How far from the center of the earth is it? Use the data in Appendix $F .(b)$ Once it reached the point found in part (a), could the spaceship turn off its engines and just hover there indefinitely? Explain.
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Step-by-Step Explanations

QUESTION

How do you calculate the gravitational force between two objects using Newton's Law of Gravitation?

STEP-BY-STEP ANSWER:

Step 1: Identify the masses of the two objects (m1 and m2).
Step 2: Measure the distance (r) between the centers of the two masses.
Step 3: Apply the formula F = G * (m1 * m2) / r^2, where G is the gravitational constant.
Step 4: Substitute the known values into the formula.
Step 5: Compute the result to find the gravitational force (F).
Final Answer: The gravitational force is given by F = G * (m1 * m2) / r^2.

Calculate Gravitational Force

QUESTION

How can Kepler's Laws be used to understand satellite motion?

STEP-BY-STEP ANSWER:

Step 1: Recognize that satellites follow elliptical orbits, as described by Kepler's First Law.
Step 2: Use Kepler's Second Law, which states that a line joining a satellite and the central body sweeps out equal areas in equal times, to assess orbital speed variations.
Step 3: Apply Kepler's Third Law to relate the orbital period to the radius of the orbit.
Step 4: Solve for the desired orbital characteristic using the established relationships.
Final Answer: Kepler's Laws allow determination of orbital shape, speed variation, and period for a satellite by relating distance and time.

Determine Satellite Orbit Characteristics

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Common Mistakes

  • Confusing gravitational potential energy with kinetic energy in gravitational fields.
  • Overlooking the inverse square nature of the gravitational force, leading to calculation errors.
  • Misinterpreting apparent weight as the true weight of an object under varying gravitational conditions.
  • Assuming Kepler's Laws apply in a straightforward manner to all celestial bodies without considering perturbative forces.