A force $\overrightarrow{\mathbf{F}}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{N}\mathbf{)}\hat{\mathbf{i}}\mathbf{+}\mathbf{c}\mathbf{}\hat{\mathbf{j}}$acts on a particle as the particle goes through displacement $\overrightarrow{\mathbf{d}}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{)}\hat{\mathbf{i}}\mathbf{-}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{)}\hat{\mathbf{j}}$. (Other forces also act on the particle.) What is c if the work done on the particle by force $\overrightarrow{\mathbf{F}}$is (a) 0 , (b) 17 J , and (c) -18 J ?

It is given that,

The force acting on a particle is $\overrightarrow{F}=4\hat{i}+c\hat{j}$

Displacement of the particle is$\overrightarrow{d}=3\hat{\mathrm{i}}-2\hat{\mathrm{j}}$

The problem deals with the work done which is the fundamental concept of physics.Work is the displacement of an object when force is applied on it. Find the value of c using the formula for work done.

Formula:

W = F . d

where,F is force, dis displacement and Wis the work done.

It is known that,

$$\begin{gathered} W=F.d \\ W=(4\hat{i}+c\hat{j})(3\hat{i}-2\hat{j}) \\ W=\left(4\right)\left(3\right)-\left(c\right)\left(2\right) \\ W=12-2c \end{gathered}$$

If the work done on the particle is zero, then,

$$\begin{gathered} 0=12-2c \\ c=6.0\mathrm{N} \end{gathered}$$

Hence, the value of c is 6.0 N, if the work done on the particle by force $\overrightarrow{F}$is zero.

If work done on the particle is 17 J, then,

$$\begin{gathered} 17=12-2\mathrm{c} \\ \mathrm{c}=-2.5\mathrm{N} \end{gathered}$$

Hence, the value of c is -2.5 N, if the work done on the particle by force$\overrightarrow{F}$is 17 J.

If work done on the particle is -18 J, then,

-18=12-2c

c = 15 N

Hence, the value of c is 15 N, if the work done on the particle by force is -18 J.