Question

The gravitational field in a region is given by vec(g)=(2hat(i)+3hat(j)) N/kg. The work done in moving a particle of mass 1 kg from (1, 1) to (2, (1)/(3)) along the line 3y + 2x = 5 is

Answer 1

Given:-

Gravitational Field line  3y +2x = 5.

Gravitational field in a region $\vec{g} = 2\hat{i} + 3\hat{j}$

To Find:-

Work Done

Solution:-

According to question gravitational line 3x + 2x = 5

For  Line slope y = $\frac{-2x + 5}{3}$

                         y = $\frac{-2x}{3} + \frac{5}{3}$

                         y = +mx +c.

                        m₁ = $\frac{-2}{3}$

Then for slope the value of m₁ = $\frac{-2}{3}$ using above value.

Now according to question, gravitational field region

                        $\vec{g} = 2\hat{i} + 3\hat{j}$

Then the gravitational field slope m₂ = $\frac{y}{x}$  = $\frac{3}{2}$.

Now using both slope value , i.e m₁xm₂ = $\frac{3}{2}x\frac{-2}{3} = -1$

Therefore the gravitational field direction is perpendicular to the line of the motion of the particle , i.e as shown in attached figure angle between m₁ and m₂ .

Then the work done W = $\vec{F}.\vec{S}$ = FScosθ

 Angle between m₁ and m₂ = $\frac{\pi }{2}$ , then W = FScos$\frac{\pi }{2}$

                                                              W = 0.