Here's how to determine the kinetic energy (K) and total energy (E) of the satellite:
1. Angular Momentum and Velocity:
The angular momentum ( J ) of a satellite in circular orbit is given by:
J=Iω=mR2ωwhere:
m is the mass of the satellite
R is the orbital radius
ω is the angular velocity
We can relate angular velocity
(ω) to linear velocity (
v ) using:
v=Rω 2. Kinetic Energy:
The kinetic energy of the satellite is:
K=‌mv2Substituting
v=Rω and
J=mR2ω, we get:
K=‌m(Rω)2=‌m(J2∕m2R4)=‌ 3. Total Energy:
The total energy ( E ) of the satellite is the sum of its kinetic energy
(K) and potential energy (U). The potential energy in a gravitational field is:
U=−‌where G is the gravitational constant and M is the mass of the Earth.
The total energy is then:
E=K+U=‌−‌We can express the gravitational constant
(G) and the mass of the Earth
(M) in terms of the acceleration due to gravity
(g) at the Earth's surface:
GM=gR2 Substituting this into the equation for total energy:
E=‌−‌=‌−gRmNow, using the centripetal acceleration equation,
g=v2∕R=(Rω)2∕R=Rω2 and substituting
J=mR2ω, we get:
gRm=Rω2m=‌Therefore, the total energy is:
E=‌−‌=−‌ Conclusion:
The correct answer is Option B:
K=‌,E=−‌