A body of mass seven kilogrammes moved 52 centimetres up the line of greatest slope of a smooth plain inclined at 60 degrees to the horizontal. Find the increase in its gravitational potential energy. Take 𝑔 to be equal to 9.8 metres per square second.
Remember, the formula we used to calculate the gravitational potential energy of a body whose mass is measured in kilograms and whose height increase is measured in metres is mass times acceleration due to gravity times ℎ. And of course, gravitational potential energy itself is measured in joules. In order to work out the increase in the height of the body, let’s draw a little sketch. We have a smooth plane inclined at 60 degrees to the horizontal. We set the starting height of the mass equal to zero. And it moves 52 centimetres up the plane.
Now, of course, we said the formula for gravitational potential energy relies on the height being measured in metres. So we divide 52 by 100, and we find that it moves 0.52 metres up the plane. We want to work out its changing height. So what we’re going to need to do is work out this measurement here. Let’s call that 𝑥 metres. And actually, we see we’ve got a right-angled triangle. So we can use right angle trigonometry to find the value of 𝑥. We know the hypotenuse. And we’re looking to find the length of the opposite site. So we’ll use the formula sin 𝜃 is equal to opposite over hypotenuse.
Here, 𝜃, the measure of the included angle, is 60 degrees. The opposite side is what we’re looking to find. We’ve called that 𝑥. And the hypotenuse is 0.52. We can solve this equation for 𝑥 by multiplying by 0.52. Now, sin 60 is root three over two. So we get 𝑥 equals 0.52 times root three over two, which we can write in fractional form as 13 root three over 50. Gravitational potential energy is therefore its mass, which is seven, multiplied by acceleration due to gravity, which is 9.8, multiplied by the increase in its height. So that’s 13 root three over 50.
We need to calculate seven times 9.8 times 13 root three over 50. That’s 30.8928 and so on, which correct to two decimal places is 30.89. We know that gravitational potential energy is measured in joules. So the increase in the gravitational potential energy of the body of mass seven kilograms is 30.89 joules.