Answer
(a) $F = -\frac{B~\pi}{L}$
(b) $F = -AL$
(c) $F = -2AL+\frac{B~\pi}{L}$
Work Step by Step
We can find an expression for $F$:
$U = Ax^2+B~sin~(\frac{\pi x}{L})$
$F = -\frac{dU}{dx} = -2Ax-\frac{B~\pi}{L}~cos~(\frac{\pi x}{L})$
(a) We can find $F$ when $x = 0$:
$F = -2Ax-\frac{B~\pi}{L}~cos~(\frac{\pi x}{L})$
$F = -2A(0)-\frac{B~\pi}{L}~cos~(0)$
$F = -\frac{B~\pi}{L}$
(b) We can find $F$ when $x = \frac{L}{2}$:
$F = -2Ax-\frac{B~\pi}{L}~cos~(\frac{\pi x}{L})$
$F = -2A(\frac{L}{2})-\frac{B~\pi}{L}~cos~(\frac{\pi}{2})$
$F = -AL$
(c) We can find $F$ when $x = L$:
$F = -2Ax-\frac{B~\pi}{L}~cos~(\frac{\pi x}{L})$
$F = -2A(L)-\frac{B~\pi}{L}~cos~(\pi)$
$F = -2AL+\frac{B~\pi}{L}$
