Question

A carpenter is applying a force $\overrightarrow{F}=-1.5x{y}^2\hat{j}$ on a hammer, whose magnitude depends on the position of the tool. The displacement of the tool from the origin to the point $(2,2)\text{m}$. Find the workdone on the hammer, if the displacement is along the straight line $y=x$.

• A. $16\text{J}$
• B. $6\text{J}$
• C. $0\text{J}$
• D. $12\text{J}$

Answer 1

The correct option is B $6\text{J}$

Given, $\overrightarrow{F}=-1.5x{y}^2\hat{j}$
Displacement $\overrightarrow{ds}=dx\hat{i}+dy\hat{j}$
But along the straight line $y=x$ , we get $\overrightarrow{ds}=dx(\hat{i}+\hat{j})$
Then, $\overrightarrow{F}.\overrightarrow{ds}=-1.5x{y}^{2}dy$

Workdone by carpenter on the tool from the origin to the point $(2,2)\text{m}$$\left(W\right)=\int \overrightarrow{F}.\overrightarrow{ds}={\int }_{y_1}^{y_2}1.5x{y}^{2}dyW=-{\int }_0^{2}1.5y^{3}dy(\because y=x)W={\left[-1.5\frac{y^4}{4}\right]}_0^{2}W=-1.5\times \frac{(2{)}^4}{4}=-6\text{J}$

Work done on the hammer is $6J$