Question

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 ?

Short answer

1. The value of cis 6.0 N, if the work done on the particle by force $\overrightarrow{F}$is zero.
2. The value of cis -2.5 N, if the work done on the particle by force $\overrightarrow{F}$is 17 J.
3. The value of cis 15 N, if the work done on the particle by force $\overrightarrow{F}$is -18 J.

Step-by-step solution

Given information

It is given that,

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

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

Determining the concept

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.

(a) Determining the value of c if the work done on the particle by force F→is zero

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.

(b) Determining the value of c if the work done on the particle by force F→is 17J

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.

(c) Determining the value of c if the work done on the particle by force F→is -18 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.