Math GRE - #31

Let $\text{F}$ be a constant force that is given by the vector $\left(\begin{array}{c}-1\\0\\1\end{array}\right)$. What is the work done by $\text{F}$ on a particle that moves along the path given by $\left(\begin{array}{c}t\\t^2\\t^3\end{array}\right)$ between time $t=0$ and $t=1$?

1. $-\dfrac{1}{4}$


2. $-\dfrac{1}{4\sqrt{2}}$


3. $0$


4. $\sqrt{2}$


5. $3\sqrt{2}$

Solution :

Choice 3 is the answer.

Recall that work can be calculated according to the line integral: $$W=\int_C{\text{F}\cdot d\text{r}}$$ along a curve $C$ with force $\text{F}$ and position vector $\text{r}$. In this case, we have $$\text{r}=\left(\begin{array}{c}t\\t^2\\t^3\end{array}\right)\implies d\text{r}=\left(\begin{array}{c}1\\2t\\3t^2\end{array}\right)dt.$$ The curve goes from 0 to 1, so our integral is: $$W=\int_0^1{\left(\begin{array}{c}-1\\0\\1\end{array}\right)\cdot \left(\begin{array}{c}1\\2t\\3t^2\end{array}\right)}dt=\int_0^1{(-1+0+3t^2)}dt=0$$ which gives zero work as we wanted.