Question

Show that for a mass distribution of constant density on a spherical surface there is no net force on a particle inside the surface.

Short answer

Answer: No, there is no net force on a particle inside the surface. The gravitational forces exerted by each infinitesimally small mass element on the spherical surface on the particle cancel out due to the symmetry of the problem.

Step-by-step solution

The Gravitational Force Formula

To begin, we need to recall the basic formula for calculating gravitational force between two masses. This is given by Newton's law of universal gravitation: F = G * (m1 * m2) / r^2 where F is the force, G is the gravitational constant (approximately 6.674 * 10^{-11} N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.

Set Up an Infinitesimally Small Mass Elemental

In order to analyze the constant density distribution on the spherical surface, consider an infinitesimally small mass (dm) on the spherical surface. Since we are looking at a constant density on the surface, dm can be expressed as: dm = σ * dA where σ is the surface mass density (mass per unit area) and dA is the infinitesimally small area element.

Consider the Gravitational Force Acting on the Particle

We can now express the gravitational force (dF) exerted by this infinitesimally small mass (dm) on the particle inside the surface as: dF = G * (dm * m) / r^2 where m is the mass of the particle inside the surface.

Express dA in Spherical Coordinates

Since we are working with a spherical surface format, it's convenient to express dA in spherical coordinates: dA = R^2 * sin(θ) dθ dφ where R is the radius of the spherical surface, θ is the polar angle, and φ is the azimuthal angle.

Express dm in Terms of dA

Earlier, we stated that dm = σ * dA. Now we can substitute the expression for dA from Step 4: dm = σ * R^2 * sin(θ) dθ dφ

Integrate Over the Entire Spherical Surface

Integrate the gravitational force, dF, exerted by each infinitesimally small mass element over the entire spherical surface to find the total net force acting on the particle inside the surface: F_net = G * m * ∫∫ (dm / r^2) However, due to the symmetry of the problem, it can be seen that the components of the forces from opposite mass elements on the spherical surface will cancel each other out about any point along the center of the sphere. Thus for each infinitesimally small mass element, there is another one that creates an equal and opposite force on the particle inside the surface.

Conclude That There is No Net Force

As the forces from these pairs of mass elements act upon the particle inside the spherical surface, the forces cancel each other out, resulting in: F_net = 0 Thus, we have demonstrated that for a mass distribution of constant density on a spherical surface, there is no net force on a particle inside the surface.

Key concepts

Newton's law of universal gravitation

Newton's law of universal gravitation is a fundamental principle that explains how two masses attract each other. The gravitational force between two objects can be calculated using the formula \( F = G \frac{m_1 m_2}{r^2} \). Here, \( F \) is the force, \( G \) is the gravitational constant (approximately \( 6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \)), \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between their centers of mass.

• It reveals that force increases with greater mass and decreases with distance.
• This law applies universally to all masses.
• Gravitational interactions play a crucial role in everything from planetary orbits to everyday objects falling to Earth.

spherical coordinates

Spherical coordinates are a set of three numbers used to locate a point in three-dimensional space. They are particularly useful for problems involving spheres, such as our current scenario of constant mass distribution on a spherical surface.
The coordinates consist of:

• \( R \): the radial distance from the origin.
• \( \theta \): the polar angle, which measures the inclination from the vertical axis.
• \( \phi \): the azimuthal angle, which describes rotation around the vertical axis.

To find an area on a spherical surface in these coordinates, we use \( dA = R^2 \sin(\theta) \, d\theta \, d\phi \). This expression helps us calculate elements of areas essential for integrating over the sphere.

surface mass density

Surface mass density \( \sigma \) is a measurement of mass distribution across a surface, defined as mass per unit area. In the context of a spherical surface, it tells us how much mass is distributed over a given area.
Considering an infinitesimally small section of the sphere, the mass \( dm \) is calculated as follows:

• \( dm = \sigma \, dA \)
• Where \( \sigma \) is constant across the sphere, indicating uniform distribution.

This uniformity simplifies calculating gravitational forces, as each small area's contribution can be easily integrated across the entire sphere to analyze the total effect.

symmetry in physics

Symmetry in physics refers to a balance or uniformity that can simplify complex problems. In this scenario, symmetry explains why there's no net gravitational force on a particle inside a spherical shell of constant density.
Due to spherical symmetry:

• Every mass element on the sphere has a counterpart that exactly opposes its gravitational pull.
• These opposing forces cancel each other out radially within the sphere.
• This cancellation results in a net force of zero, maintaining equilibrium for any particle inside.

Understanding symmetry helps in analyzing and solving problems efficiently, as it reduces complexity by revealing inherent balances.