Lesson Video: Potential Energy Mathematics

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 𝑔, 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 body’s gravitational potential energy. Consider the acceleration due to gravity 𝑔 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 body’s 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 ℎ. 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 39.2ℎ.

We can then divide both sides of this equation by 39.2, giving us a value of ℎ 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 𝑔 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 ℎ. As we are dealing with the longest side or hypotenuse and the side opposite our angle, we can use the ratio that sin 𝜃 is equal to the opposite over hypotenuse. Substituting in our values, we have sin of 30 degrees is equal to ℎ 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 ℎ 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 work–energy principle and deal with problems involving vectors. The work–energy 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 𝐴 negative six, zero to point 𝐵 negative five, four under the action of the vector force 𝐅, which is equal to 𝑚𝐢 plus two 𝐣 newtons. Given that the change in the body’s potential energy is two joules and that the displacement is in meters, determine the value of the constant 𝑚.

We are told that the body moves in a straight line from point 𝐴 to point 𝐵, where 𝐴 and 𝐵 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 𝐢 and 𝐣 in the horizontal and vertical direction, respectively, our displacement vector is equal to 𝐢 plus four 𝐣.

We are also told that the vector force acting on the body is 𝑚𝐢 plus two 𝐣 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 𝐢 plus four 𝐣 and 𝑚𝐢 plus two 𝐣.

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 𝑚 plus four multiplied by two. The 𝐢-components are one and 𝑚, and the 𝐣-components are four and two. This simplifies to 𝑚 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 𝑚 plus eight plus two must equal zero. Collecting like terms, we have 𝑚 plus 10 is equal to zero. Finally, we can subtract 10 from both sides of this equation, giving us a value of 𝑚 equal to negative 10. This means that the vector force 𝐅 is equal to negative 10𝐢 plus two 𝐣.

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 𝐅, which is equal to five 𝐢 plus three 𝐣 newtons, where 𝐢 and 𝐣 are two perpendicular unit vectors. At time 𝑡 seconds, where 𝑡 is greater than or equal to zero, the body’s position vector relative to a fixed point is given by 𝐫 is equal to 𝑡 squared plus four 𝐢 plus four 𝑡 squared plus eight 𝐣 meters. Determine the change in the body’s potential energy in the first nine seconds.

Due to the conservation of energy and the work–energy 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 𝐢 plus three 𝐣 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 𝑡 equals zero and 𝑡 equals nine.

When 𝑡 is equal to zero, we have zero squared plus four 𝐢 plus four multiplied by zero squared plus eight 𝐣. This simplifies to four 𝐢 plus eight 𝐣. When 𝑡 is equal to nine, the position vector is equal to nine squared plus four 𝐢 plus four multiplied by nine squared plus eight 𝐣. This is equal to 85𝐢 plus 332𝐣.

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

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 body’s 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.