Work Done by a General Variable Force — Question 4

Question 4

A varying vector force samples the actual curved route

A particle follows $x=t$ m and $y=t^2$ m for a dimensionless parameter $0\le t\le2$. The force is $\vec F=(2y\,\hat i+x\,\hat j)$ N, with x and y inserted in meters. Motion is prescribed by other agents.

1. Write $d\vec r$ and the force along the route.

2. Find the work of this force.

3. Why is the final force dotted with the total displacement not valid?

Question 4 - Solution

A varying vector force samples the actual curved route

(a) Parameterize both components.

The governing relation is $$\boxed{W=\int_C\vec F\cdot d\vec r=\int_0^2\vec F[\vec r(t)]\cdot\vec r^{\,\prime}(t)\,dt}$$

$$d\vec r=(1,2t)\,dt\ \mathrm m,\qquad \vec F=(2t^2,t)\ \mathrm N.$$ The parameter t is not elapsed time.

(b) Integrate the local dot product.

$$W=\int_0^2[(2t^2)(1)+t(2t)]\,dt=\int_0^2 4t^2\,dt=32/3\ \mathrm J.$$

(c) Do not replace the force history by one sample.

At the endpoint $$\vec F=(8,2)\ \mathrm N,\qquad \Delta\vec r=(2,4)\ \mathrm m,$$ whose dot product is 24 J. That calculation treats the final vector as constant everywhere, contradicting the given position-dependent force.