Ch 13: Gravitation

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 13: Gravitation

Problem 36a

Chapter 13, Problem 36a

A uniform, spherical, 1000.0-kg shell has a radius of 5.00 m. Find the gravitational force this shell exerts on a 2.00-kg point mass placed at the following distances from the center of the shell: (i) 5.01 m, (ii) 4.99 m, (iii) 2.72 m.

Verified step by step guidance

Understand the problem: We need to find the gravitational force exerted by a spherical shell on a point mass at different distances from the center of the shell. The shell has a mass of 1000.0 kg and a radius of 5.00 m, and the point mass is 2.00 kg.

Recall the shell theorem: For a point mass outside a spherical shell, the shell's gravitational force acts as if all its mass were concentrated at its center. For a point mass inside the shell, the gravitational force is zero.

For distance 5.01 m (outside the shell): Use the formula for gravitational force, $ F = \frac{G \cdot m_1 \cdot m_2}{r^2} $, where $ G $ is the gravitational constant, $ m_1 $ and $ m_2 $ are the masses, and $ r $ is the distance between the centers of the two masses. Substitute $ m_1 = 1000.0 \text{ kg} $, $ m_2 = 2.00 \text{ kg} $, and $ r = 5.01 \text{ m} $.

For distance 4.99 m (inside the shell): According to the shell theorem, the gravitational force inside a uniform spherical shell is zero. Therefore, the force on the point mass is zero.

For distance 2.72 m (inside the shell): Again, apply the shell theorem. Since the point mass is inside the shell, the gravitational force exerted by the shell is zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This law is essential for calculating the gravitational force between the shell and the point mass.

Shell Theorem

The Shell Theorem, derived from Newton's laws, states that a uniform spherical shell of matter exerts no net gravitational force on a particle located inside the shell. For a particle outside the shell, the shell's gravitational effect is as if all its mass were concentrated at its center. This theorem simplifies the calculation of gravitational forces at different distances from the shell.

Gravitational Field Inside and Outside a Spherical Shell

The gravitational field inside a spherical shell is zero due to the Shell Theorem, meaning a point mass inside experiences no gravitational force from the shell. Outside the shell, the gravitational field behaves as if the shell's mass were concentrated at its center, allowing the use of Newton's Law of Universal Gravitation to calculate the force at distances greater than the shell's radius.

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Textbook Question

A uniform, spherical, $1000.0 kg$ shell has a radius of $5.00 m$ . Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts on a point mass m as a function of the distance $r$ of $m$ from the center of the sphere. Include the region from $r equals 0$ to $r \to \infty$ .

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