Question Video: Identifying an Equation for Gravitational Potential Energy Physics

Video Transcript

The quantity ℎ is the vertical displacement of an object from a given position. The quantity 𝐸 is the object’s gravitational potential energy. Which of the following formulas correctly shows the relationship between ℎ, 𝐸, the mass of the object, and the acceleration of the object by gravity? (A) 𝐸 equals 𝑚 over 𝑔 times ℎ, (B) 𝐸 equals 𝑚 times 𝑔 times ℎ, (C) 𝐸 equals 𝑚 times 𝑔 divided by ℎ, (D) 𝐸 equals 𝑔 divided by 𝑚 times ℎ, and (E) 𝐸 equals ℎ times 𝑚 divided by 𝑔.

Let’s imagine a situation where we have ground level, and we’ll call that where ℎ is equal to zero. And that at some height we’ll call ℎ above ground level, we have an object where that object has a mass 𝑚. Because this mass is in a gravitational field whose acceleration is represented by lowercase 𝑔 pointing downward, we can describe a quantity 𝐸 which represents the gravitational potential energy of this mass. This energy depends on the mass 𝑚, and it also depends on the height ℎ and on the acceleration due to gravity 𝑔.

Notice that four out of our five answer options (A), (C), (D), and (E) all have an inverse relationship between this energy 𝐸 and one of the three variables. For example, answer choice (A) claims that 𝐸 is inversely proportional to 𝑔 and ℎ. Answer choice (D) claims that 𝐸 is inversely proportional to 𝑚 times ℎ. Regardless of the variable or variables in the denominators of these fractions, each one claims that there’s at least one of 𝑚 or 𝑔 or ℎ, which, if it increases, 𝐸 will decrease. Returning to our scenario, we can think about whether it’s likely that this is true.

For example, say that the mass 𝑚 of our object increased. What effect would this have on the gravitational potential energy of our object? Everything else being equal, a greater mass will have a greater gravitational potential energy. This means that answer option (D) is off our list. Now let’s imagine that instead of the mass of our object increasing, its height above ground level increases. If this happened, if ℎ went up, then so would the gravitational potential energy of our object. That object is now higher up in a gravitational field. This means that 𝐸 is not inversely proportional to ℎ. That eliminates answer choice (A) as well as answer choice (C).

Finally, let’s imagine that for a given height above ground level and for an object of a given mass, the acceleration due to gravity 𝑔 is increased in strength. What effect would that increase have on the object’s gravitational potential energy? Would 𝐸 tend to decrease, stay the same, or increase as a result? The greater the acceleration due to gravity 𝑔 becomes, the greater the gravitational potential energy of an object that exists in that strong field. Therefore, 𝐸 and 𝑔 are not inversely related, but rather directly related. This eliminates answer option (E) from consideration. An object’s gravitational potential energy is proportional to its mass, to its height above some reference, and to the acceleration due to gravity. We choose answer option (B).