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Question #43514

Difficulty: 8/10 | 🟢 Free

The Question

A satellite of mass $m$ is in a circular orbit of radius $R$ in the $xy$ -plane around the Earth (mass $M$ , fixed at the origin of an inertial frame). The satellite's orbital angular velocity is $\omega = \sqrt{GM/R^3}$ . At time $t=0$ , the satellite crosses the positive $x$ -axis.

A non-inertial reference frame $S'$ translates relative to the Earth. The origin of $S'$ coincides with the Earth's center at $t=0$ and moves along the positive $x$ -axis with a constant acceleration $\vec{a}_s = a_0 \hat{i}$ and zero initial velocity.

Calculate the work done strictly by the Earth's gravitational force on the satellite, as measured by an observer in frame $S'$ , during the time interval from $t=0$ to $t=\pi/\omega$ .

Options

A.
$0$
B.
$-2m R a_0$
C.
$2m R a_0$
D.
$\frac{1}{2} m a_0^2 \left(\frac{\pi}{\omega}\right)^2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Orbital Velocity Energy Balance|Frame of Reference Dependency|Non-Uniform Acceleration Constraint

Why This Question Is Tricky

Common Mistakes

Students often mistakenly assume that because gravity is a conservative central force, the work it does is simply the negative change in potential energy ( $W = -\Delta U$ ).

Since the satellite is at a distance $R$ from the Earth at both $t=0$ and $t=\pi/\omega$ , they incorrectly conclude $\Delta U = 0$ and thus $W_g' = 0$ (Option A).

However, in the accelerating non-inertial frame $S'$ , the source of the gravitational field (the Earth) is moving.

This makes the gravitational potential explicitly time-dependent in $S'$ ( $U(\vec{r}', t)$ ), meaning the gravitational force acts as a non-conservative force in this frame.

The standard $W = -\Delta U$ relation fails, and the work must be calculated via explicit path integration of $\vec{F}_g \cdot d\vec{r}'$ , accounting for the frame's relative velocity.

Option D traps students who calculate the total change in kinetic energy (which includes the work done by the pseudo-force) instead of isolating the work done by gravity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Orbital Velocity Energy Balance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free