# Does Gravity Do Work? (With Step-By-Step Examples) | Profound Physics (2023)

Gravity, to put it simply, is a force that causes objects to fall. Forces, on the other hand, have the ability to do work (change the energy of a system or convert it into other forms), so does this mean that gravity also does work?

In short, gravity does indeed do work for the simple reason that it converts potential energy into kinetic energy. It is important to realize, however, that gravity can only do work in the direction of the gravitational force, not for example, perpendicularly to the gravitational force.

The fact that gravity only does work in the direction of the force is a consequence of something called path independence, which is a property of all conservative forces (such as gravity).

I explain this idea in more detail in this article, but the bottom line is that path independence in the context of gravity means that the work done by gravity only depends on the total displacement in the direction of the gravitational force.

That probably sounds more complicated than it is, so let me explain it more.

Let’s say that some object, such as an orange (not an apple this time!) is falling downwards. The gravitational force will then be F=mg (mass of the orange times the gravitational acceleration).

The work done by gravity in the case where the orange simply falls downwards is W=mgh (h is the total height or distance the object falls). I’ll explain where this equation comes from later in the article.

Now, let’s take the orange moving down in a weird curved path that will be longer than just the downwards straight line (see picture down below). Will the work done by gravity then also be larger?

The surprising answer is actually no, the work that gravity does is still W=mgh, which means that the work done only depends on the total displacement in the direction of gravity (the height in this case), not the actual path itself.

If you didn’t know this before, the idea of path independence may seem a little weird, but quite amazingly, it is an inherent property of how gravity works.

In fact, only the work done by non-conservative forces will depend on the actual path taken (see my article on conservative vs. non-conservative forces).

Also, a very helpful and understandable explanation of this property can be found in this video from Khan Academy:

Another quite interesting thing is the concept of potential energy, which in the context of gravity, is called gravitational potential energy.

In fact, gravitational potential energy is also deeply connected to the work done by gravity (which you may have noticed above from the formula W=mgh, which is the same as the formula for gravitational potential energy, U=mgh).

Let me explain. Gravitational potential energy is actually defined in terms of the work done against gravity. Here’s the technical definition:

Gravitational potential energy is defined as the amount of work needed to move an object against the force of gravity.

Now, since the potential energy is defined as the work done against gravity, the work done and the change in potential energy will always have opposite signs. This actually gives us the definition for the work done by gravity (and also for any other conservative forces):

A nice example of the above definition is that say, an object is falling downwards due to gravity. Since its height will decrease, the change in potential energy will be negative.

Because ΔU is negative, the work done by gravity will be positive (opposite sign). The positive work will, in fact, correspond to an increase in kinetic energy as the object falls (I’ll explain this later).

(Video) Gravity, Universal Gravitation Constant - Gravitational Force Between Earth, Moon & Sun, Physics

If, on the other hand, the work done by gravity were negative, then that would correspond to a decrease in kinetic energy and an increase in potential energy (this is seen in, for example, when a ball is thrown upwards; its speed will go down and its height will increase).

The work done by gravity can also be expressed in terms of the change in kinetic energy (notice that they now have the same sign; the work will be positive if the kinetic energy increases and vice versa):

From this, we can also say that the change in kinetic energy will be the same as the change in potential energy, but with opposite sign (the increase in kinetic energy has to be the same as the decrease in potential energy):

Anyway, the key takeaways here, before we move on to some concrete examples, are the following:

• Gravity does work whenever an object moves in the direction of the gravitational force.
• The work done by gravity depends only on the total displacement in the direction of the gravitational force.
• The work done by gravity corresponds to potential energy being converted into kinetic energy or vice versa.

Quick tip: For building a deep understanding of the most important mathematical tools used in physics as well as calculus-based physics in general, I’d highly recommend my Advanced Math For Physics: A Complete Self-Study Course (link to the course homepage) as I think this is the perfect resource on these topics for someone wanting to learn advanced physics.

This course is ideal for beginners (for both self-studying and students) who want to study physics at a deeper level, but don’t know exactly where to start. You will learn everything you need to know about vectors, differential and integral calculus as well as many more advanced topics in vector calculus that not many university courses will even cover. This is done through clear and intuitive explanations, LOTS of practical examples as well as practice problems you can do yourself.

## How To Calculate The Work Done By Gravity (Step-By-Step Examples)

In this section, I’m going to show a bunch of actual practical examples of how the work done by gravity can be calculated. This is done in a step-by-step manner, but feel free to try them yourself first as exercises.

Anyway, the only formula we will really need here is the definition for work, which in the case of gravity is actually quite simple (in more complex cases, the definition for work generally involves integral calculus).

`The work done by gravity can be calculated with the formula: , where F is the gravitational force,Δr is the total displacement and θ is the angle betweenthe force and the displacement.`

From this definition, it is clear that the work done by gravity is zero in the perpendicular direction to the displacement; if the force and displacement are perpendicular, it means that θ=90° and cos(90°)=0, so the work done would also be zero.

`Quick tip: To fully understand where the above formula comes from, I'd highly recommend checking out my course on vector calculus that covers both the basics and some much more advanced stuff. You can currently get FREE access to the course here.`

Where Does The Above Formula Come From? (click to see more)

First of all, to understand the formula we will use for the examples down below, we’ll have to use the most general definition for work, which is the line integral of the dot product between a given force F and a displacement vector dr:

Now, what does this formula actually mean? Think of it this way; there is an object travelling along some given path C, along which the force may vary (which is why we have to use integration here).

The instantaneous work done by the force at a single point on the path is defined as the dot product between the force F and an infinitesimal displacement dr (this is a definition, it isn’t derived from anywhere!).

Then the total work done along the whole path C can be obtained by summing up (i.e. integrating) the instantaneous work done at each point, which is just the line integral given earlier.

Now, in the case of the force being the gravitational force, we know that the work done should only depend on the start and end points of the path (remember the path independence property).

Let’s call these points r1 and r2. Now, instead of having a path integral, we simply have an integral only between the start and end points:

In the case of gravity, if we’re reasonably close to the surface of the Earth, the gravitational force will be a constant. Next, we’ll use the definition for the dot product between two vectors, which is (see Wikipedia, for example):

(Video) Work done by gravity (path independent)| Work & Energy | Physics | Khan Academy

Inserting this into the integral above (and moving the constants F and cosθ outside the integral sign), we get:

This is, of course, the simplest possible integral to calculate, which will give us:

In the examples down below, we will indeed see exactly how the idea of path independence works in practice and also how the work done by gravity turns out to always depend on simply the total displacement (change in potential energy).

### Example: Work Done By Gravity On a Falling Object

It is known that gravity is what causes an object to fall, which means that the potential energy of the object will change. So, does gravity do work on a falling object then?

To put it simply, gravity does do work on every object that is falling due to gravity. The work done by gravity on a falling object depends only on the total change in height, given by the formula W=mgh. If an object is not falling (its height doesn’t change), gravity does not do any work on it.

The proof for this is extremely simple. Here’s a picture of the situation:

The total displacement (Δr) is simply the height h and the force will be the gravitational force F=mg. We also know that the angle θ is zero, which means that cos(0°)=1.

Inserting all of these into the formula for work (W=FΔrcosθ), we get that the work done by gravity on a falling object is W=mgh.

`Work done by gravity on a falling object: `

### Example: Work Done By Gravity On an Inclined Plane

Let’s say that there is an object placed on an inclined plane. The object will start rolling down the plane due to gravity pulling it downwards, so does gravity then do work also on an inclined plane?

In short, gravity does do work on an inclined plane, however, it will only depend on the total change in height. In other words, the work done by gravity on an inclined plane is given by the formula W=mgh, which is actually the same as the work done by gravity on a simple freely falling object.

But how exactly is that possible? Shouldn’t the work also depend on the angle of the incline or something? Well, it actually turns out that it doesn’t. Let’s look at why.

Here’s a picture of the situation:

What you’ll find after a little bit of math is that the work done by gravity will actually be the same as in the case of a simple freely falling object, W=mgh.

`Work done by gravity on an inclined plane: `

Work Done By Gravity on an Inclined Plane: Mathematical Proof (click to see more)

To find the work done by gravity on the inclined plane as shown above, we use the formula that I gave earlier:

The force F is just the gravitational force F=mg:

We can actually manipulate the above equation a little bit and see that it simplifies to a quite nice form.

From the inclined plane triangle, we can see that the angle between Δr (distance the object rolls down) and h (the total change in height of the object) is clearly θ (see picture from above) and from that, we can find the expression for cos(θ):

(Video) Astrophysicist Explains Gravity in 5 Levels of Difficulty | WIRED

From this, simply multiply by Δr and we have:

The formula for the work done then becomes:

The really interesting thing about the above formula is that the work done by gravity does not actually depend on the incline of the plane at all; it only depends on the total displacement in the direction of the gravitational force, i.e. the change in height.

### Example: Work Done By Gravity On a Pendulum

A pendulum is a system where a mass (a pendulum “bob”) is swinging back and forth under a gravitational force. Since the force of gravity is acting on the pendulum, does gravity also do work on a pendulum?

In short, gravity does do work on a pendulum for the simple reason that the height of the pendulum bob changes. The work that gravity does on a pendulum depends only on the total displacement of the pendulum bob as measured from the ground (height of the bob) and is given by the formula W=mgh.

Here is essentially what is happening; a pendulum starts swinging from some height h and we wish to calculate what the work done by gravity on this pendulum would be (everything we need is in the picture below).

Work Done By Gravity On a Pendulum: Mathematical Proof (click to see more)

The first thing we’ll do is notice that there is actually a relationship between the angles α and θ, which can be seen from the picture down below:

From this, we can get:

We can also find the displacement of the pendulum bob, which is simply the arc length as given by the picture below. For later purposes, we’ll want to use an infinitesimal displacement dr instead of the Δr we used earlier.

Since the pendulum is swinging and is generally a little bit more of a complicated system than the previous examples, the easiest method to find the work done by gravity is to simply start from the definition of work (see earlier if you don’t remember this):

Now, dr is just the arc length from the picture above (dr=Ldα). We can also use the relationship between θ and α to get another expression for the cos(θ) -term:

If you know any trigonometric relationships between sines and cosines, you’ll know that this is actually the same as simply sin(α).

Therefore, inserting the above equations (dr=Ldα and cos(θ)=sin(α) as well as the gravitational force F=mg), we get that the work done is (notice that we have to change the integration limits as well; we’re now integrating from the initial angle α to 0, where the pendulum is completely vertically):

This is a basic trigonometric integral, from which you get after evaluating the limits of integration:

We’re almost done now. We can still manipulate this equation a little bit by using the following fact from the picture:

Inserting this, we get:

Now, what is Δy – L? We know that Δy+h=L, which can be seen when the pendulum is in the vertical position. From this, we get:

If we now take the positive direction to be downwards as we’ve done with all of the previous examples as well, this then becomes the usual work done by gravity:

(Video) Gravity Explained Simply

The pendulum may seem like a complicated system, but again, the work done by gravity is just mgh. In the above proof, however, it was worth going through the calculation in explicit detail even though the result may not be too surprising at this point.

Now, the question you may have is; where does this work that the gravitational force does actually go? The answer is that it transforms into kinetic energy for the pendulum bob, which means that its velocity will increase.

How The Work Done By Gravity Makes The Pendulum Swing (click to see more)

The specific amount that the bob gains in kinetic energy will equal the work done by the gravitational force (this is just simple energy conservation).

At the bottom, all of the potential energy will have turned into kinetic energy, which means that the work done by gravity is equal to the kinetic energy of the pendulum bob (in our case, the bob will start at rest from some height h, so the initial velocity will be zero):

The work done is, as calculated earlier, simply W=mgh, so we get:

From this, we could solve what the velocity will be at the bottom:

Now, we know that a pendulum is supposed to swing back and forth dynamically, so really this kinetic energy will turn into potential energy again once the pendulum swings back up to its maximum height (the maximum height will be the same as the initial starting height!).

These dynamic changes between potential and kinetic energy then repeat (forever in an ideal case, where air resistance doesn’t exist) and this is, in fact, what causes the pendulum to swing back and forth.

### Example: Work Done By Gravity On a Satellite Orbiting Earth

When satellites are launched into space, they are put into orbit around the Earth where the gravitational force of the Earth keeps them in that orbit. Therefore, does gravity do work on a satellite orbiting the Earth?

Gravity does not do work on a satellite orbiting the Earth in a circular orbit due to the displacement of the satellite being perpendicular to the force of gravity. In an elliptical orbit, however, gravity does do work on a satellite around a small arc, but the total work done around a full orbit is zero.

First of all, let’s look at the case of a circular orbit (which is an idealized scenario; real orbits usually have more or less an elliptical shape).

In a circular orbit, the gravitational force is always pointing to the center of the Earth and the direction of the displacement is always tangential.

Therefore, the angle between the force and the displacement vectors is always 90° and the work done by gravity on the satellite orbiting in a circular orbit is zero.

Now, real orbits are not perfectly circular due to various different factors. In reality, they are elliptical. In an elliptical orbit, the direction of motion is NOT always perpendicular to the force of gravity, which means that the work done by gravity may not be zero either.

Really what this means, though, is that the work done by gravity on the satellite is non-zero during a small part of the orbit (the instantaneous work, if you will).

The total work done in a full orbit still remains zero, which ultimately has to do with the conservation of angular momentum in stable orbits, whether those be elliptical of circular.

## FAQs

### In which example is gravity doing work? ›

When I drop it, the ball travels straight downwards for 2 meters before hitting the ground. Since the direction of gravity is "down", the ball has moved in the direction of the force of gravity. Therefore, gravity has done work on the basketball! So this is the amount of work that gravity did on the basketball.

How does gravity work physics? ›

The theory states that each particle of matter attracts every other particle (for instance, the particles of "Earth" and the particles of "you") with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

In which case is gravity not doing work? ›

To put it simply, gravity does do work on every object that is falling due to gravity. The work done by gravity on a falling object depends only on the total change in height, given by the formula W=mgh. If an object is not falling (its height doesn't change), gravity does not do any work on it.

Is gravity an example of physics? ›

In physics, gravity (from Latin gravitas 'weight') is a fundamental interaction which causes mutual attraction between all things with mass or energy.

What are 5 examples of gravity? ›

Here are 10 uses for gravity that may surprise you.
• Track Earth's water and ice.
• Provide energy.
• Give a boost to spacecraft.
• Weigh the unweighable.
• Use gravity as a telescope.
• Hunt for planets around other stars.
• Investigate unseen planets.
• Save the Earth.
Sep 30, 2013

What are 3 examples of gravity? ›

Some examples of the force of gravity include:
• The force that holds the gases in the sun.
• The force that causes a ball you throw in the air to come down again.
• The force that causes a car to coast downhill even when you aren't stepping on the gas.
• The force that causes a glass you drop to fall to the floor.

What law of physics is gravity? ›

Newton's law of gravitation, statement that any particle of matter in the universe attracts any other with a force varying directly as the product of the masses and inversely as the square of the distance between them.

What is gravity in physics simple? ›

Gravity is a force that attracts a body towards the centre of the earth or any other physical body having mass.

How does gravity keep things in order? ›

The universal law of gravitation says that gravity pulls toward the centers of objects, which in this case means the ball falls in the direction of the center of the Earth. In other words, it falls downward, since “down” is always toward the center of the Earth.

Does gravity only work on falling objects? ›

A gravity force acts on an object regardless of whether it is moving or not moving. In everyday situations, the size of the gravity force on something does not change significantly as it rises above the Earth.

### Does gravity work on stationary object? ›

Even, when the object is stationary, the force of gravity acts upon it.

Does gravity act on a free falling object? ›

In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

Which is the best example of gravity? ›

One of the best examples of gravity is when a person drops a ball and it falls to the ground.

What is gravitation give an example? ›

Gravitational force is the force of attraction on a body by earth. Example - Leaves and fruits fall from a tree downwards towards the ground due to the gravitational pull.

What are 3 things gravity does? ›

Of course, we now know that gravity does far more than make things fall down. It governs the motion of planets around the Sun, holds galaxies together and determines the structure of the universe itself.

Why gravity is a force? ›

However, in the broader sense, gravity is indeed a force because it describes the resulting interaction between two masses. Gravitational effects are fundamentally caused by the warping of spacetime and the motion of objects through the warped spacetime. However, the end result is as if a force was applied.

What is an example of the law of gravity? ›

Newton proved that the force that causes, for example, an apple to fall toward the ground is the same force that causes the moon to fall around, or orbit, the Earth. This universal force also acts between the Earth and the Sun, or any other star and its satellites. Each attracts the other.

How does gravity make objects move? ›

When gravity pulls objects toward the ground, it always causes them to accelerate at a rate of 9.8 m/s2. Regardless of differences in mass, all objects accelerate at the same rate due to gravity unless air resistance affects one more than another.

How do objects move when pulled by gravity? ›

To begin answering these questions, you first need to understand that "accelerate" is the proper term, not "pull." The truth is, gravity does not "pull" objects at all; rather, gravity warps spacetime, causing objects to follow the bends that are created and, as a result, they sometimes accelerate.

What does gravity depend on? ›

The magnitude of this force depends upon the mass of each object and the distance between the centers of the two objects. Mathematically, we say the force of gravity depends directly upon the masses of the objects and inversely upon the distance between the objects squared.

### What is the law of gravity for dummies? ›

Newton's law of universal gravitation states that two bodies in space pull on each other with a force proportional to their masses and the distance between them. For large objects orbiting one another—the moon and Earth, for example—this means that they actually exert noticeable force on one another.

How does Newton's law apply to gravity? ›

Newton's law of universal gravitation states that the force of gravity affects everything with mass in the universe. Newton's law also states that the strength of gravity between any two objects depends on the masses of the objects and the distance between them.

What two things define gravity? ›

On Earth all bodies have a weight, or downward force of gravity, proportional to their mass, which Earth's mass exerts on them. Gravity is measured by the acceleration that it gives to freely falling objects.

Does gravity apply everything? ›

Every object in the universe — stars, planets, moons, even you—has gravity. Gravity is a force of attraction between all objects.

Does gravity go in all directions? ›

The effect of gravity extends from each object out into space in all directions, and for an infinite distance. However, the strength of the gravitational force reduces quickly with distance. Humans are never aware of the Sun's gravity pulling them, because the pull is so small at the distance between the Earth and Sun.

Does gravity hold everything together? ›

Gravity is what holds our world together. However, gravity isn't the same everywhere on Earth. Gravity is slightly stronger over places with more mass underground than over places with less mass. NASA uses two spacecraft to measure these variations in Earth's gravity.

Does gravity pull on an object without touching it? ›

A useful analogy for explaining the Earth's gravity force is that the Earth can pull on objects without touching them just like a magnet can affect other objects without touching them.

Does gravity work on energy? ›

Gravity is a force, so it just provides one way for objects to exchange and transform energy to different states. The kinetic energy that water gains when it falls (and can therefore be converted into electricity by a hydroelectric plant) comes ultimately from sunlight and not from gravity.

Can gravity bend anything? ›

In 1915, Albert Einstein figured out the answer when he published his theory of general relativity. The reason gravity pulls you toward the ground is that all objects with mass, like our Earth, actually bend and curve the fabric of the universe, called spacetime. That curvature is what you feel as gravity.

Does gravity work on light? ›

Yes, light is affected by gravity, but not in its speed. General Relativity (our best guess as to how the Universe works) gives two effects of gravity onlight. It can bend light (which includeseffects such as gravitational lensing), and it can change the energy oflight.

### Does gravity bend light or space? ›

Gravity bends light

Light travels through spacetime, which can be warped and curved—so light should dip and curve in the presence of massive objects. This effect is known as gravitational lensing GLOSSARY gravitational lensingThe bending of light caused by gravity .

Does gravity work at the speed of light? ›

Fomalont and Kopeikin combined observations from a series of radio telescopes across the Earth to measure the apparent change in the quasar's position as the gravitational field of Jupiter bent the passing radio waves. From that they worked out that gravity does move at the same speed as light.

Do heavier objects fall faster? ›

Acceleration of Falling Objects

Heavier things have a greater gravitational force AND heavier things have a lower acceleration. It turns out that these two effects exactly cancel to make falling objects have the same acceleration regardless of mass.

Why is there no gravity when falling? ›

When in free fall, the only force acting upon your body is the force of gravity - a non-contact force. Since the force of gravity cannot be felt without any other opposing forces, you would have no sensation of it. You would feel weightless when in a state of free fall.

Does gravity always act? ›

The gravitational force acts between all objects that have mass. This force always attracts objects together, and although it is the weakest of the four fundamental forces, gravity has an infinite range.

How is gravity important to us give examples? ›

Gravity is very important: it's why we have weight, why we naturally stay on the ground, why planes need to generate lift to get up into the air, and even why objects and planets stay in orbit!

What is meant by force of gravity give any one example of it? ›

The force of attraction on a body by earth is called gravitational force. Example : The leaves and fruits fall from a tree downwards towards the ground, water in a river flows down streams, a ball thrown up goes to a height and then returns back on ground are some examples of motion due to gravitational force.

Which is an example of a force being used to overcome gravity? ›

A rocket launches when the force of thrust pushing it upwards is greater than the weight force due to gravity downwards. This unbalanced force causes a rocket to accelerate upwards.

What is gravity examples for kids? ›

No matter how heavy or light an object is, gravity still pulls it back down to the ground. Light things are pulled down by gravity, too. Feather and ping pong balls dropped off a roof also fall down to the ground. No matter how heavy or light an object is, gravity still pulls it back down to the ground.

What is the work done by force of gravity? ›

When a satellite moves around the Earth, then the direction of force of gravity on the satellite is perpendicular to its displacement. Hence, the work done on the satellite by the Earth is zero.

### Where does the gravity force work? ›

Gravitational forces are considered to be inherently linked to what we call 'mass'. There is a gravitational force of attraction between every object in the universe. The size of the gravitational force is proportional to the masses of the objects and weakens as the distance between them increases.

Why is work done against gravity? ›

The work done against the gravity when a body is move horizontally along a frictionless surface is zero as the force of gravity acts perpendicular to the direction of motion. Was this answer helpful?

Is the work done against gravity? ›

Force of gravity acts vertically downward, while the body is move horizontally. Thus, the force of gravity is not causing the motion. So, the work done by the force of gravity is zero. Q.

Why do we say work done against gravity? ›

If the object moves upward, the work done by gravity will be negative as the direction of gravity is opposite of the displacement. Whereas if an object moves downwards, the work done by gravity will be positive as the gravity and the displacement will be in the same direction.

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