What happens to the potential energy of an object if its mass is halved while its height above the ground remains constant?

The correct option is: The potential energy is halved.

To understand why, let us look at the formula for gravitational potential energy ($PE$):

$PE=mgh$

where:
• $m$ represents the mass of the object,
• $g$ is the acceleration due to gravity (which remains constant), and
• $h$ is the height of the object above the ground.

From this equation, we can see that potential energy is directly proportional to the mass of the object ($PE\propto m$) when the height and gravity are kept constant.

If the mass is halved, the new mass ($m^{\prime }$) is:

$m^{\prime }=\frac{m}{2}$

Substituting this new mass value into the potential energy equation gives the new potential energy ($P{E}^{\prime }$):

$P{E}^{\prime }=m^{\prime }gh=\left(\frac{m}{2}\right)gh=\frac{1}{2}\left(mgh\right)=\frac{1}{2}PE$

Therefore, the potential energy of the object is halved.