5.2 Work Done in a Gravitational Field

Introduction

A gravitational field is the region of space surrounding a massive object where its gravitational influence can be felt. Any other mass placed within this field will experience an attractive force. When an object moves within this field, the gravitational force can do work on it. A key characteristic of the gravitational field is that it is a conservative field, which has important implications for the work done and the concept of potential energy.

Gravitational Field and Field Strength

• Gravitational Field: The area around a mass ($M$) where another mass ($m$) will experience a gravitational force.
• Gravitational Field Strength ($\vec{g}$): Defined as the gravitational force per unit mass at a point in the field. It is a vector quantity pointing towards the mass $M$.

$\vec{g}=\frac{{\vec{F}}_g}{m}$

The magnitude of the gravitational field strength at a distance $r$ from the center of a spherical mass $M$ is:

$g=\frac{GM}{r^2}$

Derived Units→

Conservative Fields and Conservative Forces

A field is said to be conservative if the work done by the force associated with that field depends only on the initial and final positions of the object, not on the path taken between them.

• Conservative Force: A force for which the work done is path-independent. Examples include the gravitational force, elastic spring force, and the electrostatic force.
• Non-Conservative Force: A force for which the work done does depend on the path taken. The classic example is friction or air resistance.

Two Key Properties of a Conservative Field

The path-independent nature of conservative fields leads to two defining properties:

1. Work done is independent of the path taken.
2. Work done along any closed path is zero.

Proofs of the Conservative Nature of Gravity

Proof 1: Work Done is Independent of the Path

Let's prove that the work done by gravity in moving a mass $m$ from point A to point B is the same regardless of the path taken.

Consider moving a mass $m$ from point A (at height $h_A$) to point B (at height $h_B$) in a uniform gravitational field (like near the Earth's surface).

• Path 1 (Vertical Path): Move the object straight up. The gravitational force is $F_g=mg$ (downwards) and the displacement is $\Delta h=h_B-h_A$ (upwards). The work done by gravity is:

$W_1={\vec{F}}_g\cdot \vec{d}=(mg)(h_B-h_A)\cos (180^{\circ })=-mg(h_B-h_A)$

• Path 2 (Curved Path): Now, consider any arbitrary curved path from A to B. We can break this path down into an infinite number of small horizontal and vertical steps. The gravitational force is purely vertical, so it does no work during the horizontal steps ($\cos (90^{\circ })=0$). Work is only done during the vertical steps. The sum of all the small vertical steps is equal to the total vertical displacement, $h_B-h_A$. Therefore, the total work done by gravity along the curved path is the same:

$W_2=-mg(h_B-h_A)$

Since $W_1=W_2$, the work done by gravity depends only on the initial and final heights, not on the path taken.

Proof 2: Work Done Along a Closed Path is Zero

A closed path is one where the starting point and the ending point are the same. Let's show that the work done by gravity along any closed path is zero.

Consider moving an object from point A, along some path to point B, and then back to point A along a different path.

• Work from A to B ($W_{AB}$): As shown before, this depends only on the positions of A and B. Let's say $W_{AB}=-mg(h_B-h_A)$.
• Work from B to A ($W_{BA}$): When returning, the displacement is in the opposite direction.

$W_{BA}=-mg(h_A-h_B)=mg(h_B-h_A)$

• Total Work for the Closed Path ($W_{total}$):

$W_{total}=W_{AB}+W_{BA}=-mg(h_B-h_A)+mg(h_B-h_A)=0$

Since the total work done in moving around a closed loop and returning to the starting point is zero, the gravitational field is proven to be conservative.

Possible Questions and Answers

Q: Why is friction a non-conservative force?

A: The work done by friction depends on the total distance traveled (the path length). If you slide a book in a circle on a table and return to the start, friction has done negative work over the entire path, and the total work is not zero. Therefore, friction is non-conservative.

Q: What is the significance of a field being conservative?

A: The conservative nature of a field allows us to define a potential energy associated with it. The work done by the conservative force is equal to the negative change in potential energy ($W_c=-\Delta PE$). This simplifies many problems in mechanics, allowing us to use the principle of conservation of energy.