Two identical 21.6-kg spheres of radius 14 cm are 31.0 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 find the answers to these questions, we can apply the principles of gravitational potential energy and kinetic energy.
(a) To find the speed when the spheres first make contact, we need to calculate the gravitational potential energy at the initial position and equate it to the kinetic energy at the point of contact.
Let's start by finding the initial gravitational potential energy.
The gravitational potential energy between two objects can be calculated using the formula:
PE = -G * (m1 * m2) / r
Where PE is the gravitational potential energy, G is the gravitational constant (approximately 6.67 x 10^-11 N*m^2/kg^2), m1 and m2 are the masses of the spheres, and r is the distance between their centers (31.0 cm).
PE_initial = -G * (m1 * m2) / r
Now let's find the kinetic energy at the point of contact.
Since the spheres fall toward each other, they will accelerate under the effect of gravity. The potential energy will be converted into kinetic energy as they fall closer.
We can use the principle of conservation of energy to equate the gravitational potential energy at the initial position to the kinetic energy at the point of contact:
PE_initial = KE_contact
Equating these two equations, we can solve for the velocity (v) at the point of contact:
- G * (m1 * m2) / r = (1/2) * (m1 + m2) * v^2
Solving for v^2:
v^2 = - (2 * G * (m1 * m2) / ((m1 + m2) * r))
Finally, taking the square root of both sides will give us the speed when the spheres first make contact:
v = sqrt(- (2 * G * (m1 * m2) / ((m1 + m2) * r)))
Substituting the given values:
m1 = m2 = 21.6 kg
r = 31.0 cm = 0.31 m
G = 6.67 x 10^-11 N*m^2/kg^2
v = sqrt(- (2 * (6.67 x 10^-11) * (21.6) * (21.6) / ((21.6 + 21.6) * 0.31)))
(b) To find the energy required to separate the spheres to a distance of 1.05 m, we need to calculate the change in gravitational potential energy and equate it to the work done.
The change in gravitational potential energy is given by the formula:
ΔPE = - G * (m1 * m2) * ((1/final_distance) - (1/initial_distance))
Where ΔPE is the change in gravitational potential energy, G is the gravitational constant, m1 and m2 are the masses of the spheres, final_distance is the final separation distance (1.05 m), and initial_distance is the initial separation distance (which is the sum of the radii of the spheres).
ΔPE = - G * (m1 * m2) * ((1/1.05) - (1/(2 * 0.14)))
The work done to separate the spheres is equal to the change in gravitational potential energy:
Work = ΔPE
Using the given values and solving the equation will yield the amount of energy required to separate the spheres.