Work — Question 10

Question 10

A prescribed transverse component limits possible work

For a constant force, use $W=\vec F\cdot\Delta\vec r$. A force has magnitude 10 N and fixed component $F_y=6$ N in a perpendicular planar frame. A particle displacement is $(5,0)$ m.

1. Find all possible $F_x$ values and corresponding works.

2. Can this force do 50 J over the displacement while satisfying both force conditions?

3. What transverse component would be required for 50 J if the magnitude stayed 10 N?

Question 10 - Solution

A prescribed transverse component limits possible work

(a) Use the magnitude circle.

The governing relation is $$\boxed{F_x^2+F_y^2=F^2,\qquad W=F_x\Delta x}$$

$$F_x^2+6^2=10^2,$$ so $$F_x=\pm8\ \mathrm N.$$ Work is $$5F_x=\pm40\ \mathrm J.$$

(b) Test the simultaneous constraints.

No. 50 J would require $$F_x=10\ \mathrm N,$$ inconsistent with the nonzero 6 N transverse component and total magnitude 10 N.

(c) Use the limiting aligned case.

With $$F_x=10\ \mathrm N$$ and magnitude 10 N, $F_y=0$. Maximum positive work at fixed magnitude requires the force to align with displacement.