Question

The gravitational field in a region is given by $\overrightarrow{E_g}=(4\hat{i}+\hat{j})\text{N/kg}$. Work done by this field will be zero when the particle is moved along the line

• A. $y+4x=2$
• B. $4y+x=6$
• C. $x+y=5$
• D. $x-y=5$

Answer 1

The correct option is A $y+4x=2$

Given that,

Gravitational field, $\overrightarrow{E_g}=(4\hat{i}+\hat{j})\text{N/kg}$

The gravitational force $\left(\overrightarrow{F}\right)$ on the particle will be extended along the direction of gravitational field.

Thus, workdone $\left(W\right)$ by gravitational field will be zero if $\overrightarrow{E}$ is perpendicular to the direction of displacement $\left(\overrightarrow{S}\right)$.

As, $W=\overrightarrow{F}.\overrightarrow{S}$

So, $W=0$, if $\overrightarrow{F}\perp \overrightarrow{S}$ or $\overrightarrow{E_g}\perp \overrightarrow{S}$

The slope of gravitational field line is

$\tan \theta =m_1=\frac{1}{4}$

$\Rightarrow \tan \theta =m_1=\frac{1}{4}$

Let, the slope of the direction of displacement of the particle is $m_2$.

The displacement of the particle will be perpendicular to field line only when,

$m_1{m}_2=-1..\left(1\right)$

Substitute $m_1$ in equation (1), we get

$m_2=-4$

From the equation of the straight line given in option $\left(a\right)$,

$y+4x=2$

$\Rightarrow y=-4x+2$

It's slope $=-4$

So, this is the correct equation of straight line.

Why this question ? To make the student familiar with the application of straight line equation in the concept of gravitation. Such type of problems are useful to enhance the thinking capacity of students.