### Video Transcript

In this video, we will learn how to
find gravitational potential energy and the change in it and use it to solve
different problems.

Some types of energy are easy to
visualize. For example, a fast-moving object
has more kinetic energy than a slow-moving object. The law of conservation of energy
tells us that the total energy in a system is constant. It is neither created nor
destroyed. This means that energy can only be
changed from one form to another or transferred from one object to another.

Consider the situation of a car
driving up a hill that comes to a standstill or stop due to the steepness of the
hill. What has happened to the kinetic
energy of the car? The answer is it turns into
gravitational potential energy, or GPE. This can be considered as height
energy. The higher up an object is placed,
the more gravitational potential energy it has.

The GPE of an object at any time is
equal to the mass multiplied by gravity multiplied by height. When modeling problems in this
video, we will assume that π, gravity, is equal to 9.8 meters per square
second. We will measure the height in
meters. The mass will be in kilograms. And the gravitational potential
energy will be measured in joules. For the first few questions of this
video, we will use this formula in different situations.

A crane lifts a body of mass 132
kilograms to a height of 20 meters. Find the increase in the bodyβs
gravitational potential energy. Consider the acceleration due to
gravity π equal to 9.8 meters per square second.

We are told that the crane lifts a
body of mass 132 kilograms to a height of 20 meters. We know that the gravitational
potential energy, or GPE, of a body is equal to the mass multiplied by gravity
multiplied by the height. As the gravity is equal to 9.8
meters per square second, we need to multiply 132 by 9.8 by 20. This is equal to 25,872. The increase in the bodyβs
gravitational potential energy is therefore equal to 25,872 joules.

In our next question, we will need
to calculate the height of a body when given the change in gravitational potential
energy.

A body of mass four kilograms had a
gravitational potential energy of 2,136.4 joules relative to the ground. Determine its height. Consider the acceleration due to
gravity to be 9.8 meters per square second.

We are told that the body of mass
four kilograms has a gravitational potential energy, or GPE, equal to 2,136.4
joules. We know that its acceleration due
to gravity is equal to 9.8 meters per square second. And we need to calculate the height
of the body from the ground. We know that GPE is equal to the
mass multiplied by gravity multiplied by the height. In this question, we are
multiplying four by 9.8 by β. We know this is equal to
2,136.4. Four multiplied by 9.8 is equal to
39.2, so the left-hand side becomes 39.2β.

We can then divide both sides of
this equation by 39.2, giving us a value of β equal to 54.5. The height of the four-kilogram
body with a gravitational potential energy of 2,136.4 joules is 54.5 meters.

In our next question, we will
consider a body moving up an inclined plane.

A body of mass eight kilograms
moved 238 centimeters up the line of greatest slope of a smooth plane inclined at 30
degrees to the horizontal. Calculate the increase in its
gravitational potential energy. Take π equal to 9.8 meters per
square second.

We are told that the plane is
inclined at an angle of 30 degrees and the body travels a distance of 238
centimeters. Our first step here is to convert
this into meters. As there are 100 centimeters in one
meter, 238 centimeters is equal to 2.38 meters.

We can see in our diagram that we
have a right-angled triangle. This means that we can use our trig
ratios to calculate the vertical height β. As we are dealing with the longest
side or hypotenuse and the side opposite our angle, we can use the ratio that sin π
is equal to the opposite over hypotenuse. Substituting in our values, we have
sin of 30 degrees is equal to β over 2.38. We know that sin of 30 degrees is
equal to one-half. We can then multiply both sides of
this equation by 2.38, giving us a value of β equal to 1.19. The vertical height is therefore
equal to 1.19 meters.

We are asked to calculate the
gravitational potential energy. And we know that the GPE is equal
to the mass multiplied by gravity multiplied by height. As the mass of the body was eight
kilograms and gravity is equal to 9.8 meters per square second, we need to multiply
eight, 9.8, and 1.19. Typing this into the calculator
gives us 93.296. The increase in gravitational
potential energy of the body is therefore equal to 93.296 joules.

For the remainder of this video, we
will consider the workβenergy principle and deal with problems involving
vectors. The workβenergy principle states
that the change in energy is equal to the work done on the body by the resultant
force, where the work done is equal to the force multiplied by the displacement. The force is measured in newtons
and the displacement in meters. Our work done, as with our
gravitational potential energy, is measured in joules.

When dealing with vectors, as we
will be for the remainder of this video, we can calculate the work done by finding
the dot or scalar product of the force and the displacement vectors. It is important to note that from
our conservation of energy, the work done and the change in energy must sum to
zero. The energy is only transferred and
is not created or destroyed.

A body is moving in a straight line
from point π΄ negative six, zero to point π΅ negative five, four under the action of
the vector force π
, which is equal to ππ’ plus two π£ newtons. Given that the change in the bodyβs
potential energy is two joules and that the displacement is in meters, determine the
value of the constant π.

We are told that the body moves in
a straight line from point π΄ to point π΅, where π΄ and π΅ have coordinates negative
six, zero and negative five, four. This means that we move one unit to
the right and four units up. If we consider the unit vectors π’
and π£ in the horizontal and vertical direction, respectively, our displacement
vector is equal to π’ plus four π£.

We are also told that the vector
force acting on the body is ππ’ plus two π£ newtons. We know that the work done is the
dot or scalar product of the force vector and the displacement vector. The work done is therefore equal to
the dot product of π’ plus four π£ and ππ’ plus two π£.

To calculate the dot product, we
find the sum of the products of the individual components. In this question, this is equal to
one multiplied by π plus four multiplied by two. The π’-components are one and π,
and the π£-components are four and two. This simplifies to π plus
eight.

We are also told in the question
that the change in potential energy is equal to two joules. As energy can only be transferred
and not destroyed or created, we know that the sum of the work done and the
gravitational potential energy is equal to zero. This means that π plus eight plus
two must equal zero. Collecting like terms, we have π
plus 10 is equal to zero. Finally, we can subtract 10 from
both sides of this equation, giving us a value of π equal to negative 10. This means that the vector force π
is equal to negative 10π’ plus two π£.

In our final question, we will use
vectors to find the change in potential energy over time.

A body is moving under the action
of a constant force π
, which is equal to five π’ plus three π£ newtons, where π’
and π£ are two perpendicular unit vectors. At time π‘ seconds, where π‘ is
greater than or equal to zero, the bodyβs position vector relative to a fixed point
is given by π« is equal to π‘ squared plus four π’ plus four π‘ squared plus eight
π£ meters. Determine the change in the bodyβs
potential energy in the first nine seconds.

Due to the conservation of energy
and the workβenergy principle, we know that the sum of the change in potential
energy and the work done is equal to zero. This is because energy can only be
transferred. It cannot be created or
destroyed. In this case, we are trying to
calculate the change in potential energy.

We know that work done is equal to
force multiplied by displacement. And when dealing with vectors, we
find the dot product of the force vector and displacement vector. We are told that the force is equal
to five π’ plus three π£ newtons. At present, the displacement is
unknown. We are given the position vector of
the body. And we are interested in the change
in potential energy in the first nine seconds. This means that we need to
calculate the position vector when π‘ equals zero and π‘ equals nine.

When π‘ is equal to zero, we have
zero squared plus four π’ plus four multiplied by zero squared plus eight π£. This simplifies to four π’ plus
eight π£. When π‘ is equal to nine, the
position vector is equal to nine squared plus four π’ plus four multiplied by nine
squared plus eight π£. This is equal to 85π’ plus
332π£.

We can then calculate the
displacement vector by subtracting the initial position from the final position. 85π’ minus four π’ is equal to
81π’, and 332π£ minus eight π£ is 324π£. The displacement of the body in the
first nine seconds is 81π’ plus 324π£.

We can now calculate the dot
product of the force and displacement. This is equal to the sum of five
multiplied by 81 and three multiplied by 324. This is equal to 405 plus 972,
which gives us a total work done of 1,377.

We can now use this value to
calculate the change in potential energy. As this value is positive, we know
the change in potential energy will be negative. The GPE plus 1,377 must equal
zero. This means that the change in
potential energy is negative 1,377 joules. The bodyβs potential energy has
decreased by 1,377 joules in the first nine seconds.

We will now summarize the key
points from this video. We found out in this video that the
conservation of energy means that energy can only be transferred. It cannot be created or
destroyed. This energy transfer is known as
the work done, which means that the work done plus the change in energy must equal
zero. We can calculate the work done by
multiplying the force by the displacement where the force is measured in newtons,
displacement in meters, and the work done in joules. When dealing with vectors, we find
the dot product of the force and displacement vectors.

We also found that we can calculate
the gravitational potential energy, or GPE, by multiplying the mass by the gravity
by the height. The mass is measured in kilograms,
gravity we take to be 9.8 meters per square second on Earth, and the vertical height
is measured in meters. This gives us a gravitational
potential energy measured in joules.