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Gravitational Potential Energy

What is gravitational potential energy? How does an object produce this form of energy? To answer these questions it is important to understand the meaning behind potential energy. When someone says that he or she has the potential to do great things, they're talking about something innate or hidden within the subject; the same logic applies when describing potential energy.…

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# Gravitational Potential Energy

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What is gravitational potential energy? How does an object produce this form of energy? To answer these questions it is important to understand the meaning behind potential energy. When someone says that he or she has the potential to do great things, they're talking about something innate or hidden within the subject; the same logic applies when describing potential energy. Potential energy is the energy stored in an object due to its state in a system. The potential energy could be due to electricity, gravity, or elasticity. This article goes through gravitational potential energy in detail. We will also look at the related mathematical equations and work out a few examples.

## Gravitational potential energy definition

Why does a rock dropped from a great height into a pool produce a much bigger splash than one dropped from just above the water surface? What has changed when the same rock is dropped from a greater height? When an object is elevated in a gravitational field, it gains gravitational potential energy (GPE). The elevated rock is at a higher energy state than the same rock at surface level, as more work is done to raise it to a greater height. It is called potential energy because this is a stored form of energy that when released is converted into kinetic energy as the rock falls.

Gravitational potential energy is the energy gained when an object is raised by a certain height against an external gravitational field.

The gravitational potential energy of an object depends on the height of the object, the strength of the gravitational field it is in, and the mass of the object.

If an object were to be raised to the same height from the surface of the earth or the moon, the object on the earth will have a greater GPE due to the stronger gravitational field.

The gravitational potential energy of an object increases as the height of the object increases. When the object is released and starts to fall down, its potential energy is converted into the same quantity of kinetic energy (following the conservation of energy). The total energy of the object will always be constant. On the other hand, if the object is taken to a height h work must be done, this work done will be equal to the GPE at the final height. If you calculate the potential and kinetic energies at each point when the object falls you'll see that the sum of these energies stays constant. This is called the principle of conservation of energy.

The principle of conservation of energy states that energy is neither created nor destroyed. It can however transform from one type to another.

$TE=PE+KE=\mathrm{constant}$

$\mathrm{Total}\mathrm{energy}=\mathrm{Potential}\mathrm{energy}+\mathrm{Kinetic}\mathrm{energy}=\mathrm{Constant}$

The water is stored at a height as stored potential energy. when the dam opens it releases this energy and the energy is converted into kinetic energy to drive the generators.

Water stored on top of a dam has the potential to drive hydroelectric turbines. This is because gravity is always acting on the body of water trying to bring it down. As the water flows from a height its gravitational potential energy is converted into kinetic energy. This then drives the turbines to produce electricity (electrical energy). All types of potential energy are stores of energy, which in this case is released by the opening of the dam allowing it to be converted into another form.

## Gravitational potential energy formula

The gravitational potential energy gained by an object of mass$m$when it is lifted to a height$h$in a gravitational field of$g$is given by the equation:

${E}_{GPE}\mathit{=}\mathit{}mgh$

$\mathrm{Gravitational}\mathrm{potential}\mathrm{energy}=\mathrm{mass}×\mathrm{gravitational}\mathrm{field}\mathrm{strength}×\mathrm{height}$

where${E}_{GPE}$is the gravitational potential energy in$\mathrm{joules}\left(\mathrm{J}\right)$,$m$is the mass of the object in$\mathrm{kilograms}\left(\mathrm{kg}\right)$,$h$is the height in$\mathrm{meters}\left(\mathrm{m}\right)$, and$g$is the gravitational field strength on Earth$\left(9.8\mathrm{m}/{\mathrm{s}}^{2}\right)$. But what about the work done to raise an object to a height? We already know that the increase in potential energy is equal to the work done on an object, due to the principle of conservation of energy:

${E}_{GPE}=\mathrm{work}\mathrm{done}=F×s=mgh$

$\mathrm{Change}\mathrm{in}\mathrm{gravitational}\mathrm{potential}\mathrm{energy}=\mathrm{Work}\mathrm{done}\mathrm{to}\mathrm{lift}\mathrm{the}\mathrm{object}$

This equation approximates the gravitational field as a constant, however, the gravitational potential in a radial field is given by:

$V(r)=\frac{Gm}{r}$

## Gravitational potential energy examples

Calculate the work done to raise an object of mass$5500\mathrm{g}$to a height of$200\mathrm{cm}$in the earth's gravitational field.

We know that:

$\mathrm{mass},m=5500\mathrm{g}=5.5\mathrm{kg},\phantom{\rule{0ex}{0ex}}\mathrm{height},h=200\mathrm{cm}=2\mathrm{m},\phantom{\rule{0ex}{0ex}}\mathrm{gravitaional}\mathrm{field}\mathrm{strength},g=9.8\mathrm{N}/\mathrm{kg}$

${\mathbit{E}}_{\mathbf{pe}}=mgh=5.50\mathrm{kg}x9.8\mathrm{N}/\mathrm{kg}x2\mathrm{m}=\mathbf{107}\mathbf{.}\mathbf{8}\mathbf{}\mathbf{J}$

The gravitational potential energy of the object is now$107.8\mathrm{J}$greater, which is also the amount of work done to raise the object.

Always make sure that all the units are the same as that in the formula before substituting them.

If a person weighing$75\mathrm{kg}$climbs a flight of stairs to reach a height of$100\mathrm{m}$then calculate:

(i) Their increase in${E}_{GPE}$.

(ii) The work done by the person to climb the flight of stairs.

The work done to climb the stairs is equal to the change in gravitational potential energy, Vaia Originals

First, we need to calculate the increase in gravitational potential energy when the person climbs the stairs. This can be found using the formula we discussed above.

$\begin{array}{lcl}{E}_{GPE}& =& mgh\\ & =& 75\mathrm{kg}×100\mathrm{m}×9.8\mathrm{N}/\mathrm{kg}\\ & \mathbf{=}& \mathbf{73500}\mathbf{}\mathbf{J}\mathbf{}\mathbf{or}\mathbf{}\mathbf{735}\mathbf{}\mathbf{kJ}\end{array}$

Work done to climb the stairs:

We already know that the work done is equal to the potential energy gained when the person climbs to the top of the stairs.

$\mathrm{work}=\mathrm{force}\mathrm{x}\mathrm{distance}={E}_{GPE}\mathbf{}\mathbf{=}\mathbf{}\mathbf{735}\mathbf{}\mathbf{kJ}$

The person does$735\mathrm{kJ}$work of to climb to the top of the stairs.

How many stairs would a person weighing$54\mathrm{kg}$need to climb to burn$2000\mathrm{calories}$? The height of each step is$15\mathrm{cm}$.

We first need to convert the units into the ones used in the equation.

Unit conversion:

$\begin{array}{rcl}1000\mathrm{calories}& =& 4184\mathrm{J}\\ 2000\mathrm{calories}& =& 8368\mathrm{J}\\ 15\mathrm{cm}& =& 0.15\mathrm{m}\end{array}$

First, we calculate the work done when a person climbs one step.

$mgh=54\mathrm{kg}×9.8\mathrm{N}/\mathrm{kg}×0.15\mathrm{m}\phantom{\rule{0ex}{0ex}}=\mathbf{79}\mathbf{.}\mathbf{38}\mathbf{}\mathbf{J}$

Now, we can calculate the number of steps one has to scale in order to burn$2000\mathrm{calories}$or$8368\mathrm{J}$:

$Noofsteps=\frac{8368\mathrm{J}×1000}{79.38\mathrm{J}}\phantom{\rule{0ex}{0ex}}=105,416steps$

A person weighing$54\mathrm{kg}$would have to climb$105,416\mathrm{steps}$to burn$2000\mathrm{calories}$, phew!

If a$500\mathrm{g}$apple is dropped from a height of$100\mathrm{m}$above the ground, at what speed will it hit the ground? Ignore any effects from air resistance.

The speed of a falling apple increases as it is accelerated by gravity, and is at a maximum at the point of impact, Vaia Originals

The gravitational potential energy of the object is converted into kinetic energy as it falls and increases in velocity. Therefore the potential energy at the top is equal to the kinetic energy at the bottom at the time of impact.

The total energy of the apple at all times is given by:

${\mathrm{E}}_{\mathrm{total}}={\mathrm{E}}_{\mathrm{GPE}}+{\mathrm{E}}_{\mathrm{KE}}\phantom{\rule{0ex}{0ex}}$

When the apple is at a height of$100\mathrm{m}$, the velocity is zero hence the${E}_{KE}=0$. Then the total energy is:

${E}_{\mathrm{total}}={E}_{\mathrm{GPE}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\phantom{\rule{0ex}{0ex}}$

When the apple is about to hit the ground the potential energy is zero, hence the total energy is now:

${E}_{\mathrm{total}}={E}_{\mathrm{KE}}$

Velocity during impact can be found by equating the${E}_{GPE}$to${E}_{KE}$. At the moment of impact, the kinetic energy of the object will be equal to the potential energy of the apple when it was dropped.

The apple has a velocity of$44.27\mathrm{m}/\mathrm{s}$when it hits the ground.

A small frog of mass$30\mathrm{g}$jumps over a rock of height$15\mathrm{cm}$. Calculate the change in${E}_{\mathrm{PE}}$for the frog, and the vertical speed at which the frog jumps to complete the leap.

The potential energy of a frog is constantly changing during a jump. It is zero at the moment the frog jumps and increases until the frog reaches its maximum height, where the potential energy is also maximum. After this, potential energy goes on to decrease as it is converted into kinetic energy of the falling frog. Vaia Originals

The change in energy of the frog as it makes the leap can be found as follows:

$\begin{array}{rcl}∆\mathrm{E}& =& 0.15\mathrm{m}\mathrm{x}0.03\mathrm{kg}\mathrm{x}9.8\mathrm{N}/\mathrm{kg}\\ & =& 0.0066\mathrm{J}\end{array}$

To calculate the vertical speed at take-off, we know that the total energy of the frog at all times is given by:

${E}_{\mathrm{total}}={E}_{\mathrm{GPE}}+{E}_{\mathrm{KE}}\mathbf{}\mathbf{}\phantom{\rule{0ex}{0ex}}$

When the frog is about to jump, it's potential energy is zero, hence the total energy is now

${E}_{\mathrm{total}}={E}_{\mathrm{KE}}$

When the frog is at a height of$0.15\mathrm{m}$, then the total energy is in the gravitational potential energy of the frog:

${E}_{total}={E}_{GPE}$

The vertical velocity at the start of the jump can be found by equating the${E}_{GPE}$to${E}_{KE}$.

$mgh=1/2m{v}^{2}\phantom{\rule{0ex}{0ex}}gh=1/2{v}^{2}\phantom{\rule{0ex}{0ex}}v=\sqrt{\left(2gh\right)}\phantom{\rule{0ex}{0ex}}v=\sqrt{\left(2X9.8\mathrm{N}/\mathrm{kg}X0.15\mathrm{m}\right)}\phantom{\rule{0ex}{0ex}}v=1.71\mathrm{m}/\mathrm{s}$

The frog jumps with an initial vertical velocity of$1.71\mathrm{m}/\mathrm{s}$.

## Gravitational Potential Energy - Key takeaways

• Work done to raise an object against gravity is equal to the gravitational potential energy gained by the object, measured in joules$\left(\mathrm{J}\right)$.
• Gravitational potential energy is transformed into kinetic energy when an object falls from a height.
• The potential energy is at a maximum at the highest point and it keeps reducing as the object falls.
• The potential energy is zero when the object is at ground level.
• The gravitational potential energy is given by ${E}_{GPE}=mgh$.

Gravitational potential energy is the energy gained when an object is raised by a certain height against an external gravitational field.

An apple falling from the tree, the working of a hydroelectric dam and the change in speed of a rollercoaster as it goes up and down inclines are a few examples of how gravitational potential energy is converted to velocity as the height of an object changes.

The gravitational potential energy can be calculated using Egpe=mgh

As we know, gravitational potential energy is equal to the work done to raise an object in a gravitational field. Work done is equal to force multiplied by distance (W = F x s ). This can be rewritten in terms of height, mass and gravitational field, such that h = s and F = mg. Therefore, EGPE = W = F x s = mgh.

The gravitational potential energy is given by Egpe=mgh

## Gravitational Potential Energy Quiz - Teste dein Wissen

Question

What is the energy stored in an object when it is raised against gravity called?

Gravitational potential energy

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Question

A falling object's gravitational potential energy is maximum when it reaches the ground.

False

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Question

The kinetic energy of a falling object is maximum during the moment of impact.

True

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Question

Which energy increases as an object's height is raised?

Gravitational potential energy

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The energy transferred when lifting an object to a height is equal to its ...

gravitational potential energy

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The gravitational potential energy of an object depends on ...

All of the options

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What is gravitational potential energy?

Gravitational potential energy is the energy gained when an object is raised to a certain height against an external gravitational field.

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Question

The total energy of an object changes during freefall.

False

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