Two Uniform Spheres, Each With Mass m And Radius r, Touch One Another

Introduction

In this scenario, we have two uniform spheres, each with mass m and radius r, that are touching each other. This means that the centers of the spheres are separated by a distance of 2r.

Calculating the gravitational force

To calculate the gravitational force between these two spheres, we can use the formula for the gravitational force between two masses:

F = G * (m1 * m2) / r^2

where F is the force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.

In this case, both spheres have the same mass, so we can simplify the equation to:

F = 2 * G * (m^2) / r^2

Calculating the gravitational potential energy

We can also calculate the gravitational potential energy of the system using the formula:

U = -G * (m1 * m2) / r

where U is the potential energy.

Again, since both spheres have the same mass, we can simplify the equation to:

U = -2 * G * (m^2) / r

Relationship between force and potential energy

There is a relationship between the gravitational force and potential energy of a system. Specifically, the force is the negative derivative of the potential energy with respect to distance:

F = -dU/dr

Using our equation for potential energy, we can take the derivative with respect to r:

dU/dr = 2 * G * (m^2) / r^2

And we can see that this is equal to our equation for the force between the spheres.

Equating gravitational potential energy and kinetic energy

At the point where the spheres are touching, they are not moving, so their kinetic energy is zero. This means that their total energy is equal to their potential energy. We can set the potential energy equal to the kinetic energy:

U = (1/2) * m * v^2

where v is the velocity of the spheres.

Using our equation for potential energy, we can substitute in and solve for the velocity:

-2 * G * (m^2) / r = (1/2) * m * v^2

v = sqrt( 4 * G * (m^2) / r )

Relationship between potential energy and work

Finally, we can also examine the relationship between potential energy and work. If we want to move one of the spheres away from the other, we would need to do work against the gravitational force.

The work done is equal to the change in potential energy:

W = U_final - U_initial

If we start with the spheres touching and move one of them away to a distance of d, the work done is:

W = (-2 * G * (m^2) / d) - (-2 * G * (m^2) / r)

Simplifying this, we get:

W = 2 * G * (m^2) * (1/d - 1/r)

This means that the work required to move one sphere away from the other increases as the spheres get closer together.

Conclusion

In summary, we have looked at the scenario of two uniform spheres with mass m and radius r touching each other. We calculated the gravitational force and potential energy of the system, as well as the velocity required to separate the spheres. We also examined the relationship between force and potential energy, as well as potential energy and work.