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.

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.