Two identical 21.2-kg spheres of radius 12 cm are 30.6 cm apart (center-to-center distance).

(a) If they are released from rest and allowed to fall toward one another, what is their speed when they first make contact?

(b) If the spheres are initially at rest and just touching, how much energy is required to separate them to 1.05 m apart? Assume that the only force acting on each mass is the gravitational force due to the other mass.

To solve these problems, we need to understand the concepts of gravitational potential energy and kinetic energy.

Gravitational potential energy (PE) is the energy an object possesses due to its height in a gravitational field. It is given by the equation:

PE = mgh

where m is the mass of the object, g is the acceleration due to gravity, and h is the height from a reference level.

Kinetic energy (KE) is the energy an object possesses due to its motion. It is given by the equation:

KE = (1/2)mv^2

where m is the mass of the object, and v is its velocity.

(a) To find the speed when the two spheres first make contact, we need to calculate their gravitational potential energy at the initial distance and convert it into kinetic energy at the point of contact.

1. Find the gravitational potential energy at the initial separation distance:
PE_initial = m_1 * g * h = m * g * (2r)
= m * g * (2 * 0.12 m) [r = radius = 0.12 m]

2. Convert the gravitational potential energy into kinetic energy at the point of contact:
PE_initial = KE_final
m * g * (2r) = (1/2) * m * v^2

3. Solve for v:
v^2 = 2 * g * (2r) [canceling out the mass 'm']

Now, we can plug in the known values to find the speed:
v^2 = 2 * (9.8 m/s^2) * (0.24 m)

Solving for v, we get:
v = √(2 * 9.8 * 0.24) ≈ 3.43 m/s

Therefore, the speed at the moment of contact is approximately 3.43 m/s.

(b) To find the energy required to separate the two spheres to a given distance, we need to calculate the difference in their gravitational potential energy.

1. Find the gravitational potential energy at the initial separation distance (when they are just touching):
PE_initial = m * g * (2r) = m * g * (2 * 0.12 m)

2. Find the gravitational potential energy at the final separation distance:
PE_final = m * g * h = m * g * (d - 2r) [d = final separation distance = 1.05 m]

3. Calculate the energy required to separate the spheres:
Energy required = PE_final - PE_initial

Plugging in the known values, we get:
Energy required = m * g * (d - 2r) - m * g * (2r)

Simplifying this equation, we get:
Energy required = m * g * d - 4 * m * g * r

Now, we can plug in the known values to find the energy required:
Energy required = (21.2 kg) * (9.8 m/s^2) * (1.05 m) - 4 * (21.2 kg) * (9.8 m/s^2) * (0.12 m)

Solving the equation, we get the energy required.