For an object of mass m in a circular orbit of radius R under the gravitational influence of a center of mass M, find an expression for the total energy, the sum of kinetic and potential energy (with potential energy at infinite distance set to zero), in terms of the parameters given and Newton's constant G.
Find an expression for the angular momentum L=mvR of the orbital motion (where v is the orbital velocity) in terms of m, M, R, and G.
To find the expression for the total energy, we need to consider both the kinetic energy and the potential energy.
1. Kinetic Energy (KE):
The kinetic energy is given by the formula KE = (1/2)mv^2, where m is the mass of the object and v is the orbital velocity.
2. Potential Energy (PE):
The potential energy is due to the gravitational interaction between the object of mass m and the center of mass M. Since potential energy at infinite distance is set to zero, we need to find the potential energy at the distance R. The formula for the potential energy is given by PE = -GMm/R, where G is the gravitational constant.
Now, we can find the total energy (E) by summing up the kinetic and potential energy:
E = KE + PE
= (1/2)mv^2 - GMm/R
This is the expression for the total energy of the object in a circular orbit under the gravitational influence of the center of mass.
Moving on to finding the expression for angular momentum (L), we can use the relationship between angular momentum and linear momentum.
The linear momentum (p) of the object is given by the formula p = mv, where m is the mass of the object and v is the orbital velocity.
To convert this linear momentum to angular momentum, we use the formula L = r x p, where r is the position vector from the center of mass to the object.
In a circular orbit, the position vector r is perpendicular to the linear momentum p, hence, r = R.
So, L = R x p = R x (mv)
The magnitude of the cross product is given by |A x B| = ABsinθ, where A and B are vectors, and θ is the angle between them.
Since R and p are perpendicular, sinθ = 1, then the expression for the angular momentum becomes:
L = R x (mv) = R * mv
Therefore, the expression for the angular momentum is L = m v R.
This is the expression for the angular momentum of the object in a circular orbit under the gravitational influence of the center of mass.