A uniform, solid, 1000.0-kg sphere has a radius of 5.00 m. (a) Find the gravitational force this sphere exerts on a 2.00-kg point mass placed at the following distances from the center of the sphere: (i) 5.01 m, (ii) 2.50 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 F is the gravitational force, 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 part (i), where the point mass is placed at 5.01 m from the center of the sphere, use the given values: m_1 = 1000.0 kg (mass of the sphere), m_2 = 2.00 kg (mass of the point mass), and r = 5.01 m. Substitute these values into the gravitational force formula.
3. Calculate the gravitational force for part (i) using the substituted values without solving for the numerical answer.
4. For part (ii), where the point mass is placed at 2.50 m from the center of the sphere, note that this distance is inside the sphere. Use the concept that for a uniform sphere, the gravitational force inside the sphere can be calculated by considering only the mass of the sphere that is at a radius less than or equal to the distance from the center (r = 2.50 m in this case). Calculate the effective mass of the sphere that contributes to the gravitational force at this distance.
5. Calculate the gravitational force for part (ii) using the effective mass and the distance of 2.50 m, substituting these into the gravitational force formula.

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 (F) is directly proportional to the product of the two masses (m1 and m2) and inversely proportional to the square of the distance (r) between their centers, expressed as F = G(m1*m2)/r², where G is the gravitational constant.

Point Mass

A point mass is an idealized object with mass but negligible size, allowing it to be treated as a single point in space. In gravitational calculations, point masses simplify the analysis of forces acting on them, as they do not have dimensions that would affect the gravitational interaction with other masses.

Distance from Center

The distance from the center of a mass is crucial in gravitational calculations, as it determines the strength of the gravitational force experienced by another mass. For a uniform sphere, the gravitational force behaves differently depending on whether the point mass is inside or outside the sphere, following specific rules based on the distance relative to the sphere's radius.