A body of mass eight kilograms moved 238 centimetres up the line of greatest slope of a smooth plane inclined of 30 degrees to the horizontal. Calculate the increase in its gravitational potential energy. Take 𝑔 to be equal to 9.8 metres per square second.
Let’s begin by recalling a formula we use to calculate gravitational potential energy. Gravitational potential energy, which is measured in joules, is found by multiplying the mass of the objects by acceleration due to gravity multiplied by its height. The problem is our body is not moving directly upwards. It’s moving up the line of a slope. So let’s sketch this out. The slope is inclined at 30 degrees to the horizontal. We choose the starting point of the object and let ℎ be equal to zero. The object moves up this slope at a total distance of 238 centimetres.
And of course, we want this measurement to be in metres. So we’re going to divide through by 100. So it moves up the slope 2.38 metres. The actual change in height of the object is this measurement here. Let’s call that 𝑥 metres. And then we notice we have a right-angled triangle. So we can use right angle trigonometry to calculate the value of 𝑥. We know the hypotenuse. And we’re looking to find the length of the opposite side. So we’ll use the sin ratio, sin of 𝜃 equals opposite over hypotenuse. In this case, 𝜃, the included angle, is equal to 30 degrees. And that’s equal to opposite — that’s 𝑥 — over hypotenuse — that’s 2.38. We solve this equation for 𝑥 by multiplying both sides by 2.38.
And since, of course, we know sin of 30 degrees to be equal to one-half, we see 𝑥 is equal to a half times 2.38 which is 1.19. So the change in the height of the body is 1.19 metres. Gravitational potential energy then, or the increase in the gravitational potential energy, is its mass, which is eight, multiplied by acceleration due to gravity — that’s 9.8 — multiplied by the change in height, which we found to be 1.19 metres. Eight times 9.8 times 1.19 is 93.296. And of course, we know that gravitational potential energy is measured in joules. So the increase in the gravitational potential energy of the body is 93.296 Joules.