Correct Answer - Option 4 :
The ratio of total energy of both will 4 and ratio of K.E. and P.E. will be2
CONCEPT:
Some definition of energy:
| Terms |
Gravitational Potential Energy (U) |
The kinetic energy of a satellite (KE) |
The total energy of a system (T) |
| Definition |
The gravitational potential energy of a body at a point is defined as the amount of work done in bringing the body from infinity to that point against the gravitational force.
|
The kinetic energy of a satellite in a circular orbit is half its gravitational energy and is positive instead of negative. |
The sum of the kinetic and gravitational potential energy of a system is its total energy and this total energy is conserved in orbital motion.
|
|
Mathematical expression |
\( U=-\frac{GMm}{r}\) |
\(KE=\frac{GMm}{2r}\) |
\( T=-\frac{GMm}{2r}\) |
| |
Where M = mass of earth, m = mass of the body, r = distance from the earth |
Where M = mass of earth, m = mass of the body, r = distance from the earth |
Where M = mass of earth, m = mass of the body, r = distance from the earth |
EXPLANATION:
Given - Radius of 1st satellite (r1) = R + R = 2R and Radius of 2nd satellite (r2) = R + 7R = 8R
Total energy for 1st satellite is
\( \Rightarrow T_1=\frac{-GMm}{4R}\) ----------- (1)
Total energy for 2nd satellite is
\( \Rightarrow T_2=\frac{-GMm}{16R}\) ----------- (2)
The ratio of the total energy of both satellites is
\(\Rightarrow \frac{T_1}{T_2}=\frac{\frac{-GMm}{4R}}{\frac{-GMm}{16R}}=4\)
- Therefore statement 1 is correct.
Kinetic energy for 1st satellite is
\( \Rightarrow KE_1=\frac{GMm}{4R}\) ----------- (3)
Kinetic energy for 2nd satellite is
\( \Rightarrow KE_2=\frac{GMm}{16R}\) ----------- (4)
The ratio of the kinetic energy of both satellites is
\(\Rightarrow \frac{KE_1}{KE_2}=\frac{\frac{GMm}{4R}}{\frac{GMm}{16R}}=4\)
- Therefore statement 2 is correct.
Gravitational Potential Energy for 1st satellite is
\( \Rightarrow U_1=-\frac{GMm}{2R}\) ----------- (5)
Gravitational Potential Energy for 2nd satellite is
\( \Rightarrow U_2=-\frac{GMm}{8R}\) ----------- (6)
The ratio of Gravitational Potential Energy of both satellites is
\(\Rightarrow \frac{U_1}{U_2}=\frac{\frac{-GMm}{2R}}{\frac{-GMm}{8R}}=4\)
- Therefore statement 3 is correct.
On dividing equation 3 and 5, we get
\(\Rightarrow \frac{KE_1}{U_1}=\frac{\frac{GMm}{4R}}{\frac{GMm}{2R}}=\frac{1}{2}\)
- Therefore statement 4 is incorrect.
