Suppose a large spherical object, such as a planet, with radius R and mass M has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance 𝓍 ≤ R from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius 𝓇 ≤ 𝓍 there is no net gravitational force from the mass in the spherical shell with 𝓇 > 𝓍. a. Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of x, R, m, M, and any necessary constants.

Verified step by step guidance

1. Determine the density of the spherical object. Since the object has uniform density, use the formula for density, \( \rho = \frac{M}{\frac{4}{3} \pi R^3} \), where \( M \) is the mass of the sphere and \( R \) is its radius.
2. Calculate the mass of the sphere of radius \( x \) that is influencing the particle. The mass \( m_x \) of this sphere can be found using the volume of a sphere with radius \( x \), so \( m_x = \rho \times \frac{4}{3} \pi x^3 \).
3. Apply Newton's law of universal gravitation to find the force exerted by the sphere of radius \( x \) on the particle. The gravitational force \( F \) is given by \( F = G \frac{m m_x}{x^2} \), where \( G \) is the gravitational constant.
4. Substitute the expression for \( m_x \) from step 2 into the gravitational force equation from step 3. This gives \( F = G \frac{m (\rho \times \frac{4}{3} \pi x^3)}{x^2} \).
5. Simplify the expression for the gravitational force to get the final formula. Factor out terms and simplify to get \( F = \frac{4}{3} \pi G \rho m x \). Replace \( \rho \) with its expression from step 1 to express the force entirely in terms of \( x, R, m, M, \) and \( G \).

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 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. This concept is fundamental for understanding how masses interact in a gravitational field.

Shell Theorem

The Shell Theorem states that a uniform spherical shell of mass exerts no net gravitational force on a particle located inside it. Conversely, for a particle 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 in spherical geometries, particularly when dealing with objects like planets and tunnels through them.

Uniform Density

Uniform density refers to a mass distribution where the mass per unit volume is constant throughout the object. In the context of the problem, assuming the large spherical object has uniform density allows for straightforward calculations of gravitational forces, as the mass within a radius can be easily determined using the volume formula for spheres. This assumption is crucial for applying the Shell Theorem effectively.