Question

Determine the work done by a force 

F^--> = xy î+ yz ĵ+ xzk̂

in taking a particle along 

the path defined by the equation

r(t) = tî + 2t²ĵ + t³k̂e from t=0 to t=1 . Is force conservative?

Answer 1

Given force is $\vec{F} = xy î+ yz ĵ+ xzk̂$

Since $\vec{r(}t) = tî + 2t²ĵ + t³k̂$

x is changing according to the equation $x = t$

y is changing according to the equation $y=2t^2$

z is changing according to the equation $z = t^3$

Putting value of x,y and z

Now, F will be, $\vec{F} = 2t^3 î+ 2t^5 ĵ+ t^4k̂$

$d\vec{r} = dt\hat{i}+4tdt\hat{j}+3t^2dt\hat{k}$

Then work done by the object is given by,

$W = \int_0^1 \vec{F} .d\vec{r} = \int_0^1(2t^3 î+ 2t^5 ĵ+ t^4k̂).(dt\hat{i}+4tdt\hat{j}+3t^2dt\hat{k})$

$W = \int_0^1 (2t^3+11t^6)dt = [ \frac{2}{4}t^4+\frac{11}{7}t^7 ]_0^1 = \frac{1}{2}+\frac{11}{7} = \frac{29}{14}J$