Work done by a gravitational field depends on the direction of displacement relative to the weight of the object. This work varies along different paths but remains conserved in a gravitational field.

Introduction

The gravitational field is the space around the Earth in which its gravitational pull (force) acts on a body. When an object moves within this field (either towards or away from it), work is done by the gravitational force.

Nature of Work Done by Gravitational Field

The nature of this work depends on the direction of displacement. For instance;

If the displacement is in the direction of the gravitational force, the work done is positive.
If the displacement is against the gravitational force, the work done is negative.

Work Done Along Different Paths

Consider an object of mass $m$ moving from point A to B along 3 possible paths in the presence of a gravitational force, i.e.;

Path ADB
Path ACB
Curved Path AB

The gravitational force acting on the object is equal to its weight $\left(mg\right)$ .

Path ADB

The work done along this path can be divided into two parts:

Along AD

Here,

$AD \perp \mathbf{mg}$

So,

$W_{AD} = 0$

Along DB

Here,

$AB} \parallel \mathbf{mg}$

So,

$W_{AB} = -mgh$

Total Work Done

The total work done along path ADB is;

$W_{ADB} = W_{AD} + W_{DB}$

$W_{ADB} = 0 + \left(-mgh\right)$

$W_{ADB} = -mgh$

Path ACB

The work done along this path can also be divided into two parts:

Along AC

Here,

$AC \parallel \mathbf{mg}$

So,

$W_{AC} = -mgh$

Along CB

Here,

$CB \perp \mathbf{mg}$

So,

$W_{CB} = 0$

Total Work Done

The total work done along path ACB is;

$W_{ACB} = W_{AC} + W_{CB}$

$W_{ACB} = \left(-mgh\right) + 0$

$W_{ACB} = -mgh$

Curved Path AB

The curved path can be broken into a series of small horizontal and vertical steps to find the work done.

Horizontal steps

In these steps,

$\text{weight} \perp \text{displacement}$

So,

$\text{Work Done} = 0$

Vertical steps

In these steps,

$\text{weight} \parallel \text{displacement}$

So,

$W = \sum_{i=1}^{n} \left(mg\right)_i \delta y_i$

Since the mass of the body remains the same, it means;

$W = mg \sum_{i=1}^{n} \delta y_i$

Here,

$W = -mg$ (against gravity)
$\delta y_i = \delta y_1 + \delta y_2 + \delta y_3 + … + \delta y_n = h$

Total Work Done

$W = -mg \left(\delta y_1 + \delta y_2 + … + \delta y_n\right)$

$W_{AB} = -mg h$

Inference

From all three cases, the net work done along any path from A to B is always $-mgh$ .

Work Done Along a Closed Path

The work done along a closed path in a gravitational field is zero. This property indicates that the gravitational force is conservative.

Now, let us consider a closed path such as ACBA or ADBA as shown.

Work Done by Gravitational Field along a Closed Path

$\text{work done from ADB}, W_{ADB} = -mgh, \quad \left(\text{against gravitational field}\right)$

$\text{work done from BDA}, W_{BDA} = mgh, \quad \left(\text{towards gravitational field}\right)$

$\text{work done from A to B}, W_{AB} = -mgh, \quad \left(\text{against gravitational field}\right)$

Thus, the net work done along the closed path is:

$W_{net} = W_{BDA} + W_{AB}$

$W_{net} = mgh + \left(-mgh\right)$

$W_{net} = 0$

Comparison between Conservative Force vs Non-Conservative Force

Conservative Force vs non-Conservatives Force

Friction as a Non-Conservative Force

Unlike gravitational force, frictional force is a non-conservative force. Hence, the work done against friction depends on the path followed. The reason is that friction dissipates energy as heat, making the total work path-dependent.

Conclusion

The work done by a gravitational field depends only on the initial and final positions of an object, not the path taken. This confirms that gravity is a conservative force, meaning that the total work done along a closed path is always zero.

In contrast, friction is a non-conservative force, as work done against it depends on the path taken due to energy dissipation.

Frequently Asked Questions (FAQs)

What is meant by work done by a gravitational field?

Work done by a gravitational field refers to the energy transferred when an object moves under the influence of gravity. It depends on the displacement of the object relative to the direction of the gravitational force.

How does the direction of displacement affect work done by gravity?

If the object moves upward (against gravity), the work done is negative.
If the object moves downward (in the direction of gravity), the work done is positive.
If the object moves horizontally, the work done is zero because there is no displacement in the direction of gravity.

Why is the work done the same for different paths in a gravitational field?

The work done depends only on the initial and final positions, not the path taken. This is because gravity is a conservative force, meaning the work done over a closed path is always zero.

What is the formula for work done by gravity?

The work done by gravity is given by:

$W = mgh$

Here,

$m$ = mass
$g$ = gravitational acceleration
$h$ = height difference between the initial and final positions.

Why is work done by gravity zero along a closed path?

In a closed path, an object returns to its starting position, meaning there is no net change in gravitational potential energy. As a result, the total work done is zero, confirming that gravity is a conservative force.

What is the difference between conservative and non-conservative forces?

Conservative forces (e.g., gravity) do work that depends only on initial and final positions, not the path taken.
Non-conservative forces (e.g., friction) do work that depends on the path taken, as energy is dissipated (e.g., as heat).

How does friction affect work done compared to gravity?

Friction is a non-conservative force, meaning the work done against friction depends on the length and nature of the path. Unlike gravity, which conserves mechanical energy, friction converts mechanical energy into heat.

What happens to an object’s energy when work is done by gravity?

When an object falls, gravity does positive work, increasing the kinetic energy of the object.
When an object is lifted, gravity does negative work, and energy is stored as potential energy.

Can work done by gravity ever be positive?

Yes, when an object moves downward (in the direction of gravity), the work done is positive because gravity helps in the movement.

How does work done by gravity relate to potential energy?

Work done by gravity is equal to the change in gravitational potential energy. If an object moves down by height h, the loss in potential energy equals the work done by gravity:

$W = - \Delta U$

Here, U is gravitational potential energy.

Work Done by Gravitational Field | 2 Important Types of Forces in Physics

Table of Contents

Introduction

Nature of Work Done by Gravitational Field

Work Done Along Different Paths

Path ADB

Path ACB

Curved Path AB

Inference

Work Done Along a Closed Path

Comparison between Conservative Force vs Non-Conservative Force

Friction as a Non-Conservative Force

Conclusion

Frequently Asked Questions (FAQs)

What is meant by work done by a gravitational field?

How does the direction of displacement affect work done by gravity?

Why is the work done the same for different paths in a gravitational field?

What is the formula for work done by gravity?

Why is work done by gravity zero along a closed path?

What is the difference between conservative and non-conservative forces?

How does friction affect work done compared to gravity?

What happens to an object’s energy when work is done by gravity?

Can work done by gravity ever be positive?

How does work done by gravity relate to potential energy?

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