Question

A body is displaced from prigin to (1m,1m) by force F=(2yhati + 3x^(2)hatj) along two paths (a) x=y (b) y=x^(2) Find the work done along both paths.

Answer 1

Given:

A body is displaced from origin to (1m,1m) by force F=(2yhati + 3x^(2)hatj) along two paths (a) x=y (b) y=x^(2)

To Find:

The work done along both paths.

Solution:

We know Work done = Force x displacement.

1. x = y

• dr = dxi + dyj
• dw = F.dr
• dw = ( 2yi + 3x²j).(dxi + dyj)

Since x = y ,

• dw = 2xdx + 3x²dx
• W = $\int\limits^{1,1}_{0,0} {2xdx + 3y^{2} dy} \,$  = x² + y³ | = 1 + 1 = 2J

     2. y = x²

• dr = dxi + dyj

• dw = F.dr
• dw = ( 2yi + 3x²j).(dxi + dyj)

Since x² = y ,

• dw = 2x²dx + 3ydy
• W = $\int\limits^{1,1}_{0,0} {2x^{2} dx + 3ydy} \,$  = $\frac{2x^{3} }{3}$ + $\frac{3y^{2}}{2}$ | = 2/3 + 3/2 = 13/6 J