Work — Question 6

Question 6

An angular bound becomes a work bound

A constant pull has magnitude in $[8,10]$ N and angle in $[30^\circ,60^\circ]$ to a 3 m displacement. All combinations are allowed and the displacement is exact.

1. Find the smallest and largest possible work.

2. Could the work be 28 J under these bounds?

3. If only the magnitude bound were known, what full work interval would be possible over all directions?

Question 6 - Solution

An angular bound becomes a work bound

(a) Use the monotonic cosine on this interval.

The governing relation is $$\boxed{W=Fd\cos\theta,\qquad\cos60^\circ\leq\cos\theta\leq\cos30^\circ}$$

$$W_{\min}=8(3)\cos60^\circ=12\ \mathrm J.$$

$$W_{\max}=10(3)\cos30^\circ=15\sqrt3\approx25.981\ \mathrm J.$$

(b) Compare against the extreme.

No. 28 J is larger than the maximum allowed with these angular bounds, although it is below the unrestricted magnitude bound of 30 J.

(c) Allow opposing as well as aligned directions.

Over all directions, $$-Fd\le W\le Fd.$$ The union over the allowed magnitudes is $[-30,30]\ \mathrm J$. A direction restriction contains information that a force magnitude alone does not.