Question of Physics

Question

i) Derive a relation between a conservative force (F(x; y; z)) and the potential energy associated with this force U(x; y; z).

ii) What is the result of the following integral, explain your answer briefly

∫F_c.dr

where

F_c and dr represent the conservative force and the infinitesimal displacement vector, respectively, and the integral is a line integral along a closed path.

iii) What can you say about the sign of work done by a friction force?

iv) What can you say about the result of the following integral:

∫F_fr.dr

where

F_fr represents a friction force?

Answer 1

So for all forces - conservative and non-conservative - we have the relationship U = -∫ F•dr.

For conservative forces, however, the result of this integral is path-independent, and therefore you can choose to evaluate the integral along any path you desire, so long as it starts and ends at the original starting/end points.

So, if you are evaluating a conservative force over a closed interval, you can choose to evaluate over a 'zero-length' path going immediately from the start to the endpoint, evaluating to zero.

Since you have presumably seen this before, I'll leave 3 & 4 to you - just think about the change in energy of an object being affected by friction.