WEBVTT
00:00:00.640 --> 00:00:11.320
In this video, we will learn how to find gravitational potential energy and the change in it and use it to solve different problems.
00:00:12.960 --> 00:00:16.400
Some types of energy are easy to visualize.
00:00:16.960 --> 00:00:24.880
For example, a fast-moving object has more kinetic energy than a slow-moving object.
00:00:25.640 --> 00:00:31.840
The law of conservation of energy tells us that the total energy in a system is constant.
00:00:32.680 --> 00:00:35.760
It is neither created nor destroyed.
00:00:36.800 --> 00:00:44.960
This means that energy can only be changed from one form to another or transferred from one object to another.
00:00:45.720 --> 00:00:54.240
Consider the situation of a car driving up a hill that comes to a standstill or stop due to the steepness of the hill.
00:00:54.920 --> 00:00:57.800
What has happened to the kinetic energy of the car?
00:00:59.000 --> 00:01:04.360
The answer is it turns into gravitational potential energy, or GPE.
00:01:05.120 --> 00:01:08.520
This can be considered as height energy.
00:01:09.040 --> 00:01:14.280
The higher up an object is placed, the more gravitational potential energy it has.
00:01:15.400 --> 00:01:24.920
The GPE of an object at any time is equal to the mass multiplied by gravity multiplied by height.
00:01:26.120 --> 00:01:38.640
When modeling problems in this video, we will assume that π, gravity, is equal to 9.8 meters per square second.
00:01:39.680 --> 00:01:41.800
We will measure the height in meters.
00:01:42.120 --> 00:01:44.120
The mass will be in kilograms.
00:01:44.680 --> 00:01:49.080
And the gravitational potential energy will be measured in joules.
00:01:50.240 --> 00:01:57.960
For the first few questions of this video, we will use this formula in different situations.
00:01:59.480 --> 00:02:09.280
A crane lifts a body of mass 132 kilograms to a height of 20 meters.
00:02:10.000 --> 00:02:15.000
Find the increase in the bodyβs gravitational potential energy.
00:02:16.160 --> 00:02:25.680
Consider the acceleration due to gravity π equal to 9.8 meters per square second.
00:02:26.520 --> 00:02:37.040
We are told that the crane lifts a body of mass 132 kilograms to a height of 20 meters.
00:02:37.600 --> 00:02:48.520
We know that the gravitational potential energy, or GPE, of a body is equal to the mass multiplied by gravity multiplied by the height.
00:02:49.440 --> 00:03:04.520
As the gravity is equal to 9.8 meters per square second, we need to multiply 132 by 9.8 by 20.
00:03:05.200 --> 00:03:10.800
This is equal to 25,872.
00:03:11.480 --> 00:03:21.760
The increase in the bodyβs gravitational potential energy is therefore equal to 25,872 joules.
00:03:22.320 --> 00:03:30.000
In our next question, we will need to calculate the height of a body when given the change in gravitational potential energy.
00:03:31.000 --> 00:03:44.680
A body of mass four kilograms had a gravitational potential energy of 2,136.4 joules relative to the ground.
00:03:45.440 --> 00:03:46.600
Determine its height.
00:03:47.720 --> 00:03:55.600
Consider the acceleration due to gravity to be 9.8 meters per square second.
00:03:56.680 --> 00:04:10.760
We are told that the body of mass four kilograms has a gravitational potential energy, or GPE, equal to 2,136.4 joules.
00:04:12.000 --> 00:04:21.440
We know that its acceleration due to gravity is equal to 9.8 meters per square second.
00:04:22.560 --> 00:04:25.400
And we need to calculate the height of the body from the ground.
00:04:25.960 --> 00:04:35.440
We know that GPE is equal to the mass multiplied by gravity multiplied by the height.
00:04:36.720 --> 00:04:42.960
In this question, we are multiplying four by 9.8 by β.
00:04:43.600 --> 00:04:49.960
We know this is equal to 2,136.4.
00:04:49.960 --> 00:05:02.440
Four multiplied by 9.8 is equal to 39.2, so the left-hand side becomes 39.2β.
00:05:03.560 --> 00:05:17.000
We can then divide both sides of this equation by 39.2, giving us a value of β equal to 54.5.
00:05:18.000 --> 00:05:35.720
The height of the four-kilogram body with a gravitational potential energy of 2,136.4 joules is 54.5 meters.
00:05:36.720 --> 00:05:41.200
In our next question, we will consider a body moving up an inclined plane.
00:05:42.440 --> 00:05:57.160
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.
00:05:57.880 --> 00:06:01.120
Calculate the increase in its gravitational potential energy.
00:06:02.000 --> 00:06:08.040
Take π equal to 9.8 meters per square second.
00:06:08.920 --> 00:06:19.880
We are told that the plane is inclined at an angle of 30 degrees and the body travels a distance of 238 centimeters.
00:06:20.760 --> 00:06:26.320
Our first step here is to convert this into meters.
00:06:27.680 --> 00:06:39.920
As there are 100 centimeters in one meter, 238 centimeters is equal to 2.38 meters.
00:06:40.800 --> 00:06:45.920
We can see in our diagram that we have a right-angled triangle.
00:06:46.680 --> 00:06:53.200
This means that we can use our trig ratios to calculate the vertical height β.
00:06:54.120 --> 00:07:07.080
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.
00:07:07.800 --> 00:07:17.920
Substituting in our values, we have sin of 30 degrees is equal to β over 2.38.
00:07:19.120 --> 00:07:22.760
We know that sin of 30 degrees is equal to one-half.
00:07:23.400 --> 00:07:35.880
We can then multiply both sides of this equation by 2.38, giving us a value of β equal to 1.19.
00:07:35.880 --> 00:07:43.680
The vertical height is therefore equal to 1.19 meters.
00:07:44.400 --> 00:07:47.640
We are asked to calculate the gravitational potential energy.
00:07:48.280 --> 00:07:57.200
And we know that the GPE is equal to the mass multiplied by gravity multiplied by height.
00:07:57.880 --> 00:08:16.200
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.
00:08:16.200 --> 00:08:24.600
Typing this into the calculator gives us 93.296.
00:08:24.600 --> 00:08:35.520
The increase in gravitational potential energy of the body is therefore equal to 93.296 joules.
00:08:36.360 --> 00:08:45.760
For the remainder of this video, we will consider the workβenergy principle and deal with problems involving vectors.
00:08:46.480 --> 00:09:02.040
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.
00:09:02.600 --> 00:09:06.080
The force is measured in newtons and the displacement in meters.
00:09:06.760 --> 00:09:12.640
Our work done, as with our gravitational potential energy, is measured in joules.
00:09:13.320 --> 00:09:29.720
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.
00:09:30.840 --> 00:09:41.480
It is important to note that from our conservation of energy, the work done and the change in energy must sum to zero.
00:09:42.600 --> 00:09:47.920
The energy is only transferred and is not created or destroyed.
00:09:48.680 --> 00:10:05.800
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.
00:10:06.520 --> 00:10:15.960
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.
00:10:16.880 --> 00:10:31.800
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.
00:10:31.800 --> 00:10:38.920
This means that we move one unit to the right and four units up.
00:10:39.680 --> 00:10:53.160
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£.
00:10:53.960 --> 00:11:02.800
We are also told that the vector force acting on the body is ππ’ plus two π£ newtons.
00:11:03.360 --> 00:11:12.560
We know that the work done is the dot or scalar product of the force vector and the displacement vector.
00:11:13.600 --> 00:11:23.840
The work done is therefore equal to the dot product of π’ plus four π£ and ππ’ plus two π£.
00:11:24.680 --> 00:11:31.480
To calculate the dot product, we find the sum of the products of the individual components.
00:11:32.240 --> 00:11:39.360
In this question, this is equal to one multiplied by π plus four multiplied by two.
00:11:40.200 --> 00:11:47.040
The π’-components are one and π, and the π£-components are four and two.
00:11:47.840 --> 00:11:51.320
This simplifies to π plus eight.
00:11:52.160 --> 00:11:57.280
We are also told in the question that the change in potential energy is equal to two joules.
00:11:58.120 --> 00:12:11.640
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.
00:12:12.800 --> 00:12:18.880
This means that π plus eight plus two must equal zero.
00:12:20.160 --> 00:12:26.560
Collecting like terms, we have π plus 10 is equal to zero.
00:12:27.640 --> 00:12:35.360
Finally, we can subtract 10 from both sides of this equation, giving us a value of π equal to negative 10.
00:12:36.200 --> 00:12:43.840
This means that the vector force π
is equal to negative 10π’ plus two π£.
00:12:44.200 --> 00:12:50.920
In our final question, we will use vectors to find the change in potential energy over time.
00:12:51.640 --> 00:13:04.560
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.
00:13:05.680 --> 00:13:26.080
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.
00:13:27.480 --> 00:13:31.880
Determine the change in the bodyβs potential energy in the first nine seconds.
00:13:32.720 --> 00:13:42.920
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.
00:13:43.920 --> 00:13:46.880
This is because energy can only be transferred.
00:13:47.440 --> 00:13:50.920
It cannot be created or destroyed.
00:13:51.360 --> 00:13:55.800
In this case, we are trying to calculate the change in potential energy.
00:13:56.600 --> 00:14:01.840
We know that work done is equal to force multiplied by displacement.
00:14:02.800 --> 00:14:11.040
And when dealing with vectors, we find the dot product of the force vector and displacement vector.
00:14:12.200 --> 00:14:17.920
We are told that the force is equal to five π’ plus three π£ newtons.
00:14:19.120 --> 00:14:21.520
At present, the displacement is unknown.
00:14:21.960 --> 00:14:24.280
We are given the position vector of the body.
00:14:25.080 --> 00:14:32.000
And we are interested in the change in potential energy in the first nine seconds.
00:14:32.000 --> 00:14:39.080
This means that we need to calculate the position vector when π‘ equals zero and π‘ equals nine.
00:14:40.440 --> 00:14:51.760
When π‘ is equal to zero, we have zero squared plus four π’ plus four multiplied by zero squared plus eight π£.
00:14:52.640 --> 00:14:58.280
This simplifies to four π’ plus eight π£.
00:14:59.280 --> 00:15:12.200
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£.
00:15:13.200 --> 00:15:20.480
This is equal to 85π’ plus 332π£.
00:15:21.360 --> 00:15:28.120
We can then calculate the displacement vector by subtracting the initial position from the final position.
00:15:29.480 --> 00:15:46.080
85π’ minus four π’ is equal to 81π’, and 332π£ minus eight π£ is 324π£.
00:15:46.640 --> 00:15:56.040
The displacement of the body in the first nine seconds is 81π’ plus 324π£.
00:15:57.440 --> 00:16:02.400
We can now calculate the dot product of the force and displacement.
00:16:03.600 --> 00:16:12.880
This is equal to the sum of five multiplied by 81 and three multiplied by 324.
00:16:14.080 --> 00:16:28.200
This is equal to 405 plus 972, which gives us a total work done of 1,377.
00:16:29.440 --> 00:16:34.200
We can now use this value to calculate the change in potential energy.
00:16:34.600 --> 00:16:41.840
As this value is positive, we know the change in potential energy will be negative.
00:16:42.960 --> 00:16:49.880
The GPE plus 1,377 must equal zero.
00:16:51.120 --> 00:16:58.080
This means that the change in potential energy is negative 1,377 joules.
00:16:59.440 --> 00:17:07.040
The bodyβs potential energy has decreased by 1,377 joules in the first nine seconds.
00:17:08.360 --> 00:17:13.000
We will now summarize the key points from this video.
00:17:14.040 --> 00:17:20.520
We found out in this video that the conservation of energy means that energy can only be transferred.
00:17:20.960 --> 00:17:24.000
It cannot be created or destroyed.
00:17:24.640 --> 00:17:34.520
This energy transfer is known as the work done, which means that the work done plus the change in energy must equal zero.
00:17:35.440 --> 00:17:46.400
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.
00:17:47.520 --> 00:17:54.280
When dealing with vectors, we find the dot product of the force and displacement vectors.
00:17:55.200 --> 00:18:04.680
We also found that we can calculate the gravitational potential energy, or GPE, by multiplying the mass by the gravity by the height.
00:18:05.400 --> 00:18:19.680
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.
00:18:20.240 --> 00:18:24.880
This gives us a gravitational potential energy measured in joules.