Video: Gravitational Potential Energy

An object held at a point 1.5 m above the ground has 1,176 J of gravitational potential energy. What is the mass of the object?

02:36

Video Transcript

An object held at a point 1.5 meters above the ground has 1176 joules of gravitational potential energy. What is the mass of the object?

All right, in this example, let’s say that this is our ground level and that this shape here is our object. We’re told that this object is 1.5 meters above the ground and that its gravitational potential energy is equal to 1176 joules. Given this information, we want to solve for the mass of this object. To do this, we can recall a mathematical relationship connecting mass, height, and gravitational potential energy. This relationship says that the gravitational potential energy of an object is equal to its mass multiplied by the acceleration due to gravity that the object experiences times its height above some minimum height value. In our case, it’s not GPE we want to solve for though, but it’s the object’s mass 𝑚.

To help us do that, we can divide both sides of this equation by 𝑔 times ℎ. That way, on the right-hand side, a factor of 𝑔 cancels in the numerator and denominator, and so does a factor of ℎ. So then, an object’s mass is equal to its gravitational potential energy divided by 𝑔 times ℎ. Now, in our case, where we have an object that’s 1.5 meters above ground, we can treat the acceleration due to gravity 𝑔 as exactly 9.8 meters per second squared. So, our object’s mass 𝑚 is equal to its gravitational potential energy — which we’re given, that’s 1176 joules — divided by 𝑔 and by ℎ. And for our scenario, ℎ, the height of our object, is 1.5 meters. With all these values plugged into our equation, before we go ahead and calculate 𝑚, let’s work for a bit with the units in these values.

Looking at our denominator, we see we have units of meters per second squared times meters. If we collect these units on the right-hand side of our denominator, what we find is we have meters squared per second squared. And now let’s look at the units in the numerator, joules. A joule is equal to a newton times a meter. And a newton itself is equal to a kilogram meter per second squared. This means a newton meter is a kilogram meter per second squared times meters, or a kilogram meter squared per second squared. But notice now that meter squared per second squared appears in our numerator and our denominator. Therefore, when we calculate this fraction, those parts of our units cancel out. This means that the final unit we’ll end up with is kilograms. And that’s just as it should be since we’re calculating a mass. And when we go ahead and calculate 𝑚, we find a result of 80 kilograms. That is the mass of our object.

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