You release a block from the top of a long, slippery inclined plane of length \(\ell\) that makes an angle \(\theta\) with the horizontal. The magnitude of the block's acceleration is \(g \sin \theta\).

(a) For an \(x\) axis pointing down the incline and having its origin at the release position, derive an expression for the potential energy of the block-Earth system as a function of \(x\).

(b) How would your expression change if you put the origin of the \(x\) axis at the bottom of the incline?

(c) Does the location of the origin make any difference in the dynamics of the situation?

(d) Use the expression you derived in part \(a\) to determine the speed of the block at the bottom of the incline. Does your speed value agree with the result obtained by a kinematic calculation?