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. Step 1: Begin by understanding the problem. The gravitational force on the particle inside the tunnel is due to the mass of the sphere with radius 𝓇 ≤ 𝓍. The mass outside this radius does not contribute to the net gravitational force. This is a result of the shell theorem, which states that the gravitational force inside a spherical shell is zero.
2. Step 2: Calculate the density of the spherical object. Since the object has uniform density, the density \( \rho \) can be expressed as \( \rho = \frac{M}{\frac{4}{3} \pi R^3} \), where \( M \) is the total mass of the sphere and \( R \) is its radius.
3. Step 3: Determine the mass of the sphere with radius \( \mathcal{r} \leq \mathcal{x} \). The mass \( M_{\text{inside}} \) of the sphere with radius \( \mathcal{x} \) is given by \( M_{\text{inside}} = \rho \cdot \frac{4}{3} \pi \mathcal{x}^3 \). Substituting \( \rho \), this becomes \( M_{\text{inside}} = \frac{M}{R^3} \cdot \mathcal{x}^3 \).
4. Step 4: Use Newton's law of gravitation to find the gravitational force. The gravitational force \( F \) on the particle of mass \( m \) is given by \( F = \frac{G \cdot m \cdot M_{\text{inside}}}{\mathcal{x}^2} \), where \( G \) is the gravitational constant. Substituting \( M_{\text{inside}} \), this becomes \( F = \frac{G \cdot m \cdot \frac{M}{R^3} \cdot \mathcal{x}^3}{\mathcal{x}^2} \).
5. Step 5: Simplify the expression for the gravitational force. After simplifying, the force becomes \( F = \frac{G \cdot m \cdot M \cdot \mathcal{x}}{R^3} \). This is the final expression for the gravitational force on the particle inside the tunnel.

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. This means that when considering a particle within a spherical object, only the mass that is at a radius less than the particle's distance from the center contributes to the gravitational force acting on it. This theorem simplifies the analysis of gravitational forces in spherical objects.

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 force, as the mass within a given radius can be easily determined using the volume formula for spheres. This assumption is crucial for deriving the expression for gravitational force on the particle.