Work Done by Gravitational Force

Introduction

Gravity is one of the most familiar and fundamental forces in nature. It governs how objects fall, how planets orbit the Sun, and even how galaxies are bound together. While we often experience gravity as a constant downward pull, physicists describe its effects more rigorously using the concept of work done by gravitational force.

Whenever a body moves under the influence of gravity, gravity either does work on it or has work done against it. This concept is not only crucial in classical mechanics but also forms the backbone of topics like potential energy, planetary motion, and energy conservation.

In this article, we will dive deeply into:

The meaning of work in physics.
The definition of gravitational force.
The derivation of work done by gravity.
Different cases (vertical, inclined, orbital).
The relationship with potential energy.
Real-life applications, misconceptions, and solved examples.

By the end, you will understand how this “invisible hand” of nature performs measurable work that shapes our universe.

Work in Physics: A Quick Refresher

In physics, work is not just effort—it has a precise mathematical definition. W=F⃗⋅d⃗=Fdcos⁡θW = \vec{F} \cdot \vec{d} = F d \cos \thetaW=F⋅d=Fdcosθ

Where:

WWW = work done by the force
FFF = magnitude of the force applied
ddd = displacement of the object
θ\thetaθ = angle between force and displacement

Key points:

Work is scalar (it has magnitude but no direction).
Work is positive when force and displacement are in the same direction.
Work is negative when force and displacement are opposite.
If there is no displacement, or the force is perpendicular to motion, work is zero.

Gravitational Force

Gravity is a universal attractive force acting between two masses.

Newton’s Law of Gravitation

F=Gm1m2r2F = G \frac{m_1 m_2}{r^2}F=Gr2m1m2

Where:

GGG = universal gravitational constant (6.67×10−11 Nm2/kg26.67 \times 10^{-11} \, Nm^2/kg^26.67×10−11Nm2/kg2)
m1,m2m_1, m_2m1,m2 = masses of objects
rrr = distance between centers of masses

Near Earth’s Surface

For small heights compared to Earth’s radius, gravitational force on a mass mmm is: F=mgF = mgF=mg

where g≈9.8 m/s2g \approx 9.8 \, m/s^2g≈9.8m/s2.

This approximation simplifies our calculations in everyday mechanics.

Work Done by Gravitational Force (Derivation)

Case 1: Object Moving Vertically

Consider a mass mmm lifted vertically upward through a height hhh.

Force of gravity: F=mgF = mgF=mg (downward).
Displacement: d=hd = hd=h (upward).
Angle between them: θ=180∘\theta = 180^\circθ=180∘. W=Fdcos⁡θ=(mg)(h)(cos⁡180∘)=−mghW = F d \cos \theta = (mg)(h)(\cos 180^\circ) = -mghW=Fdcosθ=(mg)(h)(cos180∘)=−mgh

So, work done by gravity is negative when lifting upward.

Now, if the object falls downward through height hhh: W=(mg)(h)(cos⁡0∘)=+mghW = (mg)(h)(\cos 0^\circ) = +mghW=(mg)(h)(cos0∘)=+mgh

So, work done by gravity is positive during free fall.

Case 2: Object on an Inclined Plane

If an object slides down a frictionless incline of length LLL and height hhh:

Gravity acts vertically downward.
Only the vertical component of displacement contributes to work.

W=mghW = mghW=mgh

Thus, work done by gravity depends only on vertical height, not on the path.

Case 3: General Path Independence

If an object moves from point A at height h1h_1h1 to point B at height h2h_2h2, work done by gravity is: W=mg(h1−h2)W = mg(h_1 – h_2)W=mg(h1−h2)

This shows that gravitational work is path independent—it depends only on initial and final positions, not on the actual path taken.

Case 4: Work in Orbital Motion

In orbital mechanics, when a satellite moves in a circular orbit around Earth, displacement is always perpendicular to gravitational force. θ=90∘⇒W=0\theta = 90^\circ \quad \Rightarrow \quad W = 0θ=90∘⇒W=0

Thus, in a perfectly circular orbit, gravity does no work.
But when the orbit changes (elliptical motion, escape, etc.), work is done.

Relation to Potential Energy

Work done by gravity is directly related to change in gravitational potential energy (U). U=mghU = mghU=mgh

Change in potential energy when moving from height h1h_1h1 to h2h_2h2: ΔU=mg(h2−h1)\Delta U = mg(h_2 – h_1)ΔU=mg(h2−h1)

Since work and energy are linked: Wgravity=−ΔUW_{gravity} = – \Delta UWgravity=−ΔU

This is the work-energy relation for conservative forces.

Graphical Interpretation

Work vs. Height:
- Work done by gravity is proportional to change in height.
Potential Energy Curve:
- As height increases, potential energy increases.
- Gravity does negative work while lifting.

Real-Life Examples of Work Done by Gravity

Free Falling Object
- Gravity does positive work, converting potential energy into kinetic.
Throwing a Ball Upward
- Gravity does negative work, reducing the ball’s kinetic energy.
Water in a Dam
- Stored water at height has gravitational potential energy; as it flows down, gravity does positive work to generate electricity.
Roller Coasters
- As cars descend slopes, gravity does positive work, increasing speed.
Climbing Hills
- Work must be done against gravity; gravity itself does negative work on the climber.
Orbiting Satellites
- In circular orbits, gravity does no work; in elliptical orbits, it alternately does positive and negative work.

Advanced Case: Work with Universal Gravitation

For large distances (beyond Earth’s surface), gravitational force varies with distance.

Work done moving a mass mmm from distance r1r_1r1 to r2r_2r2: W=∫r1r2GMmr2 drW = \int_{r_1}^{r_2} \frac{GMm}{r^2} \, drW=∫r1r2r2GMmdr W=GMm(1r1−1r2)W = GMm \left( \frac{1}{r_1} – \frac{1}{r_2} \right)W=GMm(r11−r21)

This is used in escape velocity, satellite launching, and astrophysics.

Key Properties of Work Done by Gravity

Conservative Force:
- Work depends only on initial and final points, not on the path.
Sign of Work:
- Positive when moving downward.
- Negative when moving upward.
Zero Work:
- If displacement is horizontal (like walking on a flat road), gravity does no work.

Common Misconceptions

Gravity always does work.
- False. If displacement is horizontal, work is zero.
Work by gravity is always negative.
- Not true. It’s positive when displacement is downward.
Potential energy is always positive.
- Incorrect. In universal gravitation, potential energy is often negative.
Path affects work.
- Wrong. Only height difference matters.

Historical Background

Aristotle believed heavier objects fall faster, ignoring energy considerations.
Galileo proved all masses fall equally and highlighted motion under gravity.
Newton formalized universal gravitation, enabling the precise calculation of work and energy relationships.

Solved Problems

Problem 1

A 10 kg object falls from a height of 20 m. Calculate the work done by gravity.

Solution: W=mgh=(10)(9.8)(20)=1960 JW = mgh = (10)(9.8)(20) = 1960 \, JW=mgh=(10)(9.8)(20)=1960J

Problem 2

A person lifts a 5 kg bag from the ground to a shelf 2 m high. Work done by gravity? W=−mgh=−(5)(9.8)(2)=−98 JW = -mgh = -(5)(9.8)(2) = -98 \, JW=−mgh=−(5)(9.8)(2)=−98J

Negative because displacement is upward.

Problem 3

A satellite of mass 500 kg moves from orbit radius 8000 km8000 \, km8000km to 12000 km12000 \, km12000km. Find work done by gravity. W=GMm(1r1−1r2)W = GMm \left(\frac{1}{r_1} – \frac{1}{r_2}\right)W=GMm(r11−r21)

Insert values and solve → (students can calculate).

Problem 4

On a hill of height 100 m, a 50 kg cart slides down. Work done by gravity? W=mgh=(50)(9.8)(100)=49,000 JW = mgh = (50)(9.8)(100) = 49,000 \, JW=mgh=(50)(9.8)(100)=49,000J

Practice Questions

Derive work done by gravity for motion along an inclined plane.
Show that gravity is a conservative force using work-energy relations.
Why does gravity do no work in circular orbit?
Calculate the work done when raising a 70 kg person up 5 floors (each 3 m high).
A ball thrown upward with velocity 20 m/s comes back to the ground. Show total work done by gravity is zero.

Applications in Daily Life & Technology

Sports: High jump, basketball shots, ski slopes—all rely on gravity’s work.
Hydroelectric Power: Conversion of gravitational potential to electrical energy.
Engineering: Designing bridges, roller coasters, elevators requires careful energy calculations.
Space Science: Rocket launching, escape velocity, satellite orbits depend on gravitational work.