How Does an Additive Constant Impact Potential Energy Calculations?

[shanepitts]

Pictured below was a problem shown in class with solution. I didn't have time to ask the professor a question about the last step involving an additive constant.

V is potential energy, r_e=Earth radius and z is distance from Earth's surface.

What is an additive constant, and how does it allow the last transformation?

 

[nasu]

You know (I hope) that the value of the potential energy depends on the reference point.
In the first formula, the reference point is at infinite. That means the energy becomes closer and closer to zero as z increases to infinite.

The other formula, U=mgz uses the surface of the Earth as reference.
So you need to change your formula to take this into account. Without the additive term, the energy at z=0 will be -mgR.
By adding +mgR you make the energy zero at z=0.

 

[DEvens]

An additive constant is a constant that you add on. A very typical example is when you do an indefinite integral.

$\int x dx = \frac{x^2}{2} + c$

Here $c$ is an additive constant.

The zero of a potential can be defined where you like. This is because you only ever see differences in potential from one place to another. (Well... I guess strictly speaking that is not universally true. But for purposes of your homework assignment it is.) That means you can add any constant onto the potential and get an equally valid potential. All it does is redefine where the zero is. By using the form that your assignment has it has defined the zero of $V(z)$ to be at $z=0$ which seems a natural place to define it.

 

[shanepitts]

You know (I hope) that the value of the potential energy depends on the reference point.
In the first formula, the reference point is at infinite. That means the energy becomes closer and closer to zero as z increases to infinite.

The other formula, U=mgz uses the surface of the Earth as reference.
So you need to change your formula to take this into account. Without the additive term, the energy at z=0 will be -mgR.
By adding +mgR you make the energy zero at z=0.

Thank you for your detailed answer, I fully fathom now.

 

[shanepitts]

An additive constant is a constant that you add on. A very typical example is when you do an indefinite integral.

$\int x dx = \frac{x^2}{2} + c$

Here $c$ is an additive constant.

The zero of a potential can be defined where you like. This is because you only ever see differences in potential from one place to another. (Well... I guess strictly speaking that is not universally true. But for purposes of your homework assignment it is.) That means you can add any constant onto the potential and get an equally valid potential. All it does is redefine where the zero is. By using the form that your assignment has it has defined the zero of $V(z)$ to be at $z=0$ which seems a natural place to define it.

Thanks