A uniform, spherical, 1000.0-kg shell has a radius of 5.00 m. (a) 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

1. Identify the relevant formula for gravitational force, which is given by Newton's law of universal gravitation: F = G \frac{m_1 m_2}{r^2}, where G is the gravitational constant (6.67430 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}), m_1 and m_2 are the masses of the two objects, and r is the distance between the centers of the two masses.
2. For the case (i) where the point mass is at 5.01 m from the center of the shell, use the shell theorem which states that the gravitational force outside a uniform spherical shell of mass acts as if all the mass were concentrated at the center. Calculate the force using the formula with r = 5.01 m.
3. For the case (ii) where the point mass is at 4.99 m from the center of the shell, use the shell theorem which states that the gravitational force inside a uniform spherical shell is zero. Therefore, the gravitational force exerted on the point mass is zero.
4. For the case (iii) where the point mass is at 2.72 m from the center of the shell, apply the same reasoning as in case (ii) since the point mass is still inside the spherical shell. The gravitational force exerted on the point mass remains zero.
5. Summarize the results: For distances outside the shell, calculate the force using the gravitational formula, and for distances inside the shell, the gravitational force is zero.

Key Concepts

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. This force can be calculated using the formula F = G(m1*m2)/r^2, where G is the gravitational constant.

Shell Theorem

The Shell Theorem states that a uniform spherical shell of mass exerts no net gravitational force on any mass located inside it. For a mass outside the shell, the shell can be treated as if all its mass were concentrated at its center. This theorem simplifies the calculation of gravitational forces for spherical objects.

Distance and Gravitational Interaction

The distance between two masses significantly affects the gravitational interaction between them. As the distance increases, the gravitational force decreases according to the inverse square law. In this problem, the distances from the center of the shell to the point mass determine whether the mass is inside, on the surface, or outside the shell, influencing the gravitational force experienced.