Problem 68 - A Complete Resource Book in Physics for JEE Main

Question

The time period of an earth satellite in circular orbit is independent of (A) the mass of the satellite. (B) radius of its orbit. (C) both the mass and radius of the orbit. (D) neither the mass of the satellite nor the radius of its orbit.

Step-by-Step Solution

Answer

The time period of an Earth satellite in a circular orbit is independent of (A) the mass of the satellite.

Step 1: Gravitational force acting on the satellite

The gravitational force acting on a satellite in orbit can be represented as: \(F_g = G \frac{M_Em_s}{r^2}\) Where: - \(F_g\) = gravitational force between the Earth and the satellite - \(G\) = gravitational constant - \(M_E\) = mass of the Earth - \(m_s\) = mass of the satellite - \(r\) = distance between the center of the Earth and the satellite (radius of the orbit)

Step 2: Centripetal force acting on the satellite

The centripetal force required for the satellite to maintain its circular path is given by: \(F_c = m_s \frac{v^2}{r}\) Where: - \(F_c\) = centripetal force - \(v\) = orbital speed of the satellite

Step 3: Equate gravitational force and centripetal force

Since the centripetal force needed by the satellite to remain in orbit is provided by the gravitational force, we can equate the two equations: \(G \frac{M_Em_s}{r^2} = m_s \frac{v^2}{r}\)

Step 4: Solve for orbital speed

We can now solve for the orbital speed of the satellite: \(v = \sqrt{G\frac{M_E}{r}}\)

Step 5: Derive the formula for the time period of a satellite

The time period (T) of the satellite is given by the ratio of the circumference of the orbit (2πr) to the orbital speed (v): \(T = \frac{2 \pi r}{v}\) Plugging the value of \(v\) from Step 4 into the equation: \(T = \frac{2 \pi r}{\sqrt{G\frac{M_E}{r}}}\)

Step 6: Simplify the equation for the time period

We can now simplify the equation for the time period as: \(T = 2 \pi \sqrt{\frac{r^3}{GM_E}}\)

Step 7: Analyze the given options

From the derived formula, we can see that the time period (T) of the satellite is independent of: - the mass of the satellite (m_s) -- Option (A) And depends on: - the radius of its orbit (r) -- Option (B) Therefore, the correct answer is: The time period of an Earth satellite in a circular orbit is independent of (A) the mass of the satellite.