Physics

Problem

A uniform gravitational field

\boldsymbol{g}= -g\hat{j}

(downward) acts in a region of space. A particle of mass

m

is moved from point

A(x_{1},y_{1},z_{1})

to point

B(x_{2},y_{2},z_{2})

along an arbitrary smooth curve described by the vector function

\boldsymbol{r}(s)

, where

s

is the path length. a. (a) Express the infinitesimal work

\mathrm{d}W

done by the gravitational force during an element

\mathrm{d}\boldsymbol{r}

of the path. b. (b) Evaluate the total work

W_{AB}

as a line integral and show that it simplifies to

mg(y_{1}-y_{2})

. c. (c) What conclusion can you draw about the dependence of gravitational work on the actual path followed? d. (d) For

m=2\,\text{kg}

y_{2}-y_{1}=5\,\text{m}

and

g=9.8\,\text{m}\,\text{s}^{-2}

, calculate

W_{AB}