Question: A particle moves along a straight path through displacement $\overrightarrow{d}\mathbf{=}\left(\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{m}\right)\hat{i}\mathbf{+}\mathbf{c}\hat{j}$while force$\overrightarrow{F}\mathbf{=}\left(\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{N}\right)i\mathbf{+}\left(\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{N}\right)j$ acts on it. (Other forces also act on the particle.) What is the value of c if the work done by $\overrightarrow{\mathbf{F}}$on the particle is (a) zero, (b) positive, and (c) negative?

It is given that,

Force$\overrightarrow{F}=\left(2\text{N}\right)\hat{\text{i}}-\left(4\text{N}\right)\hat{\text{j}}$

Distance$\overrightarrow{d}=\left(8\text{m}\right)\hat{\text{i}}-\left(c\right)\hat{\text{j}}$

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.Using the formula of work done, find the value offrom the given force and distance and find if the value of c is zero, positive, or negative.

Formula:

$W=\overrightarrow{F}\overrightarrow{d}$

Where,

F is force; dis displacement and W is the work done.

Now,

$W=\overrightarrow{F}.\overrightarrow{d}$

As,

role="math" localid="1661002427804" $$\begin{gathered} \overrightarrow{F}=\left(2\text{N}\right)\hat{\text{i}}-\left(4\text{N}\right)\hat{\text{j}} \\ \overrightarrow{d}=\left(8\text{m}\right)\hat{\text{i}}-\left(c\right)\hat{\text{j}} \\ W=\left[\left(2\text{N}\right)\hat{\text{i}}-\left(4\text{N}\right)\hat{\text{j}}\right].\left[\left(8\text{m}\right)\hat{\text{i}}-\left(c\right)\hat{\text{j}}\right] \\ \text{W=16-4c} \end{gathered}$$

If W = 0

0 = 16 - 4C

C = 4 m

Therefore, the value of c = 4m if work done by$\overrightarrow{F}$on the particle is zero.

Now,

$W=\overrightarrow{F}.\overrightarrow{d}$

As

$$\begin{gathered} \overrightarrow{F}=\left(2\text{N}\right)\hat{\text{i}}-\left(4\text{N}\right)\hat{\text{j}} \\ \overrightarrow{d}=\left(8\text{m}\right)\hat{\text{i}}-\left(c\right)\hat{\text{j}} \\ W=\left[\left(2\text{N}\right)\hat{\text{i}}-\left(4\text{N}\right)\hat{\text{j}}\right].\left[\left(8\text{m}\right)\hat{\text{i}}-\left(c\right)\hat{\text{j}}\right] \\ \text{W=16-4c} \end{gathered}$$

If W > 0 then 16 > 4C

Therefore,

C> 4

Therefore, the value of c if work done by $\overrightarrow{F}$on the particle is positive.

Now,

$W=\overrightarrow{F}.\overrightarrow{d}$

As,

$$\begin{gathered} \overrightarrow{F}=\left(2\text{N}\right)\hat{\text{i}}-\left(4\text{N}\right)\hat{\text{j}} \\ \overrightarrow{d}=\left(8\text{m}\right)\hat{\text{i}}-\left(c\right)\hat{\text{j}} \end{gathered}$$

$W=\left[\left(2\text{N}\right)\hat{\text{i}}-\left(4\text{N}\right)\hat{\text{j}}\right].\left[\left(8\text{m}\right)\hat{\text{i}}-\left(c\right)\hat{\text{j}}\right]$

W= 16-4c

If W < 0 then 16 < 4c

Therefore,

C> 4m

Therefore, the value of c if work done by$\overrightarrow{F}$on the particle is negative.