Work — Question 5

Question 5

Work measurements can recover force components

For a constant force, use $W=\vec F\cdot\Delta\vec r$. An unknown constant planar force does 12 J during displacement $(3,0)$ m and 5 J during displacement $(1,1)$ m.

1. Find $F_x$ and $F_y$.

2. Predict its work during displacement $(-2,4)$ m.

3. If only the first measurement were available, what family of forces would remain possible?

Question 5 - Solution

Work measurements can recover force components

(a) Use two independent equations.

The governing relation is $$\boxed{W_j=F_x\Delta x_j+F_y\Delta y_j}$$ $3F_x=12$ gives $$F_x=4\ \mathrm N.$$ Then $$F_x+F_y=5$$ gives $$F_y=1\ \mathrm N.$$

(b) Project onto the new displacement.

$$W=4(-2)+1(4)=-4\ \mathrm J.$$ Work can be negative even with both force components positive.

(c) Identify the unmeasured perpendicular component.

Only $$F_x=4\ \mathrm N$$ would be fixed; every real $F_y$ would remain possible. One work measurement does not determine a full planar force vector.