Q74P - Fundamentals Of Physics

Question

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?

Step-by-Step Solution

Answer

Answers

 

1. The value of c = 4m if work done by$\overrightarrow{F}$  on the particle is zero.
2. The value of c < 4m if work done by $\overrightarrow{F}$ on the particle is positive.

The value of c > 4m  if work done by $\overrightarrow{F}$ on the particle is negative

Step 1: Given information

It is given that,

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

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

Step 2: 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.Using the formula of work done, find the value of from 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; d is displacement and W is the work done.

Step 3: (a) determining the value of c if work done by F &#8594; 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

0 = 16 - 4C

C = 4 m

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

 

Step 4: (b) determining the value of c if work done by F &#8594; on the particle is positive

Now,

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

As

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

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.

Step 5: (c) Determining the value of c if work done by F &#8594; on the particle is negative

 

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.