Video Transcript
In this video, we will learn how to
determine the potential energy for an object due to the gravitational force acting
on it. We know from experience that when
we jump up in the air, we don’t just keep moving upwards forever. Instead, we jump up to some maximum
height before falling back down towards the surface of the Earth. Eventually, we land back on the
ground that we jumped from. The reason for this is the
gravitational force exerted on us by Earth. In fact, Earth also exerts a
gravitational force on all of the objects around us. This force attracts objects towards
the center of the Earth, causing objects to accelerate towards the ground.
For example, imagine holding a ball
some height above the ground. At first, we are holding the ball
in place. This means that it is stationary,
with zero speed. However, if we let go of the ball,
we know that it will fall towards the ground. This is because of the
gravitational force of Earth. When the ball is falling, it is
clearly no longer stationary. It now has some speed. The gravitational force of the
Earth causes the ball to accelerate. As the ball’s speed increases, its
kinetic energy increases. Remember that kinetic energy is the
category of energy associated with an object’s motion. Now, a very important thing to
remember about energy is that it’s a conserved quantity. That means that energy cannot be
created or destroyed; it can only be transferred.
In the case of the falling ball
then, we know that this increasing kinetic energy can’t just come out of
nowhere. In order for the ball’s kinetic
energy to increase as it falls, the energy in some other category must decrease. We call this category of energy
gravitational potential energy, which often gets shortened to simply potential
energy. It’s worth pointing out, by the
way, that there are more types of potential energy than just gravitational. So strictly speaking, the term
“potential energy” applies more generally. However, for the purposes of this
video, if we mention potential energy, then we will always be referring to
gravitational potential energy.
Now, gravitational potential energy
is the category of energy associated with the height of an object above the surface
of the Earth when a gravitational force is exerted on it. The gravitational potential energy
of an object depends on three things: the mass of the object, the object’s height
above Earth’s surface, and the strength of the gravitational force that is exerted
on it, which is indicated by a quantity called the acceleration due to gravity. The greater any of these three
quantities, the greater the potential energy.
Let’s start by thinking about the
gravitational force and the acceleration due to gravity. We should note that a greater
acceleration due to gravity corresponds to a stronger gravitational force. The acceleration due to gravity at
the Earth’s surface is often denoted as a lowercase 𝑔. The value of this quantity 𝑔 is
approximately equal to 9.8 meters per second squared. This is also sometimes rounded to
10 meters per second squared. Then, the other two factors that
affect the potential energy are the object’s mass and its height above the
ground. The gravitational potential energy
of an object is directly proportional to both its mass and its height.
Suppose that we have two objects,
both at the same height above the surface of the Earth, one with a mass of one
kilogram and the other twice this mass, that is, a mass of two kilograms. Since gravitational potential
energy is proportional to mass, the two-kilogram object must has twice the potential
energy of the one-kilogram object because it has twice the mass. That is, doubling the mass of an
object doubles its gravitational potential energy.
Now let’s suppose that we instead
have two objects with the same mass of one kilogram. In this case, one has a height of
one meter above the ground, while the other has a height of two meters that’s twice
the height of the one-meter-high object. Since gravitational potential
energy is also proportional to height, then the object that is two meters above the
ground must has twice the potential energy of the object that is one meter above the
ground. So, we’ve seen that the
gravitational potential energy of an object depends on these three factors: its
mass, its height above the ground, and the value of the acceleration due to
gravity. We can also describe mathematically
how these factors affect the potential energy using an equation.
Specifically, this equation says
that an object’s gravitational potential energy is equal to its mass multiplied by
the acceleration due to gravity multiplied by the object’s height above Earth’s
surface. Let’s abbreviate gravitational
potential energy as GPE. We can also rewrite the right-hand
side of the equation symbolically as 𝑚 times 𝑔 times ℎ. Here, 𝑚 is the object’s mass, 𝑔
is the acceleration due to gravity near the surface of Earth, and ℎ is the object’s
height above the ground. When we measure the mass in units
of kilograms, the acceleration due to gravity in meters per second squared, and the
height in meters, the gravitational potential energy will have units of joules,
written as a capital J.
Let’s now see how we can use this
formula in order to calculate the potential energy of an example object. We’ll consider a book that’s at
rest on a shelf. Let’s say that this book has a mass
𝑚 of 0.2 kilograms and that the shelf holds the book at a height ℎ of two meters
above the ground. Remember that we also know that the
acceleration due to gravity at Earth’s surface, 𝑔, is approximately equal to 9.8
meters per second squared. We can substitute these three
values into this formula to calculate the book’s gravitational potential energy. We have that this is equal to 0.2
kilograms multiplied by 9.8 meters per second squared multiplied by two meters.
With these units for the quantities
on the right-hand side, we’ll calculate an energy in joules. Evaluating the expression, we get a
result of 3.92 joules. This value is the gravitational
potential energy of the book, that is, the energy that the book has as a result of
its height while the Earth is exerting a gravitational force on it.
Now, we said earlier that the
gravitational force on an object is indicated by this quantity called the
acceleration due to gravity. This gravitational force is equal
to the weight of the object. We’ll denote the weight with a
capital 𝑊. The weight is related to the
acceleration due to gravity via this equation here. The weight, or gravitational force,
is equal to the object’s mass 𝑚 multiplied by the acceleration due to gravity
𝑔. Since weight is a force, it can be
measured in units of newtons, denoted capital N. Specifically, we’ll calculate a
weight in newtons so long as we have a mass with units of kilograms and an
acceleration due to gravity in meters per second squared.
Looking again at our expression for
the gravitational potential energy, we can see that it contains the quantity 𝑚
multiplied by the quantity 𝑔. we can recognize this as the right-hand side of our
expression for the weight 𝑊. We know then that 𝑚 multiplied by
𝑔 is equal to 𝑊. This means we can replace, or
substitute, 𝑊 in place of 𝑚 times 𝑔 in the formula for gravitational potential
energy. We have that the gravitational
potential energy of an object is equal to the object’s weight 𝑊 multiplied by the
object’s height ℎ. This formula provides another way
of calculating the gravitational potential energy of an object if we know the weight
and the height.
Let’s now take a look at a couple
of example questions.
A ball moves slowly when at the top
of a slope. At the bottom of the slope, the
speed of the ball is much greater. What category of energy decreased
as the kinetic energy of the ball increased? (A) Elastic potential energy, (B)
electrical potential energy, (C) gravitational potential energy.
Okay, so in this question, we’re
given a diagram showing a ball rolling down a slope. We’re told that the speed of the
ball is much greater at the bottom of the slope than it was at the top. So between this position and this
position, the speed of the ball increases. We are told in the question that
the kinetic energy of the ball increases. Now, this makes sense because
kinetic energy is the category of energy associated with the motion of an
object. So if the speed of an object
increases, the object’s kinetic energy must also increase.
We’re asked what category of energy
decreased as the kinetic energy increased. Now, it’s important to recall here
that energy is a quantity that is always conserved. This means that energy cannot be
created or destroyed; it can only be transferred. As the ball’s kinetic energy
increases then, its energy in another category must decrease. We’re being asked which of these
three categories of energy is the one that decreases.
Elastic potential energy is related
to the shape of the object. We’re not told anything about the
shape of the ball changing, so there’s no reason for its elastic potential energy to
change. Electrical potential energy is
related to the electric charge of an object. Again, there is no reason why this
would change for the ball in this question. Gravitational potential energy
depends on the height of an object above Earth. Since the ball rolls down a slope,
we know the height of the ball decreases. Then, since the height decreases,
this means that the ball’s gravitational potential energy also decreases. This gravitational potential energy
is transferred to the kinetic energy that the ball gains. We can therefore identify the
correct answer as option (C). The category of energy that
decreased as the ball’s kinetic energy increased is gravitational potential
energy.
Okay, now let’s have a look at one
more question.
An object with a mass of two
kilograms is 17 meters above the ground. Find the gravitational potential
energy for the object to the nearest joule. Use 9.8 meters per second squared
for the acceleration due to gravity.
We’re being asked here to find the
gravitational potential energy for an object that has a mass of two kilograms and a
height of 17 meters above the ground. Let’s suppose that this here is our
object. We know that its mass, which we’ve
labeled as 𝑚, is two kilograms. We also know that the height of
this object above the ground, which we’ve labeled as ℎ, is 17 meters. Now, we don’t know how the object
is at this height. For example, it could be at rest on
a shelf, or it could be midflight. However, in this case, the
specifics don’t matter. We’re just being asked about the
object’s gravitational potential energy.
Let’s recall that gravitational
potential energy, or GPE for short, is the category of energy associated with the
height of an object above the surface of the Earth. Let’s also recall that the
gravitational potential energy of an object is equal to the object’s mass 𝑚
multiplied by 𝑔, the acceleration due to gravity, multiplied by the height ℎ of the
object above the Earth’s surface. We know the values of 𝑚 and ℎ in
this case, and we’re told to use 9.8 meters per second squared for the acceleration
due to gravity 𝑔.
We can now go ahead and substitute
our values for the quantities 𝑚, 𝑔, and ℎ into our equation for the gravitational
potential energy. We find that the object’s
gravitational potential energy is equal to two kilograms multiplied by 9.8 meters
per second squared multiplied by 17 meters. With a mass in units of kilograms,
an acceleration due to gravity in meters per second squared, and a height in meters,
we’ll calculate an energy value in units of joules. When we evaluate this expression,
we get a result of 333.2 joules. We should note though that we’re
asked to give our answer to the nearest joule. Rounding this value to the nearest
joule gives a result of 333 joules. Our answer then is that, to the
nearest joule, the gravitational potential energy of the object is 333 joules.
Let’s now finish up by summarizing
what we have learnt in this video.
We began by learning that
gravitational potential energy is the category of energy associated with the height
of an object above the surface of the Earth. We saw that the gravitational
potential energy, or GPE, of an object can be calculated using the formula 𝑚 times
𝑔 times ℎ, where 𝑚 is the mass of the object, 𝑔 is the acceleration due to
gravity, and ℎ is the object’s height above the Earth’s surface.
We learnt that at the Earth’s
surface, the value of the acceleration due to gravity is approximately 9.8 meters
per second squared. Then, we saw that the weight of an
object, which is a force, is equal to the object’s mass 𝑚 multiplied by the
acceleration due to gravity 𝑔. Finally, we saw how we could
rewrite the gravitational potential energy equation using the quantity weight, as
gravitational potential energy is equal to weight 𝑊 multiplied by height ℎ.
