Calculate the work done by a variable force= − − 2F ,ax bx where a and b areconstants, in moving a particle of mass mfrom point 1x x= to 2x x= along astraight line

To calculate the work done by a variable force, we need to integrate the force with respect to displacement. In this case, the force is given by F = -2F(ax + bx), where a and b are constants.

First, let's find the displacement between point 1x and 2x. The displacement is given by Δx = 2x - 1x = x.

Next, we can substitute the force equation into the work formula: W = ∫F dx.

Substituting the force equation, we have W = ∫(-2F(ax + bx)) dx.

Now, we can distribute the -2F term: W = -2F ∫(ax + bx) dx.

Integrating each term separately, we get W = -2F ∫ax dx + -2F ∫bx dx.

Integrating, we have W = -2F (a/2)x^2 + -2F (b/2)x^2.

Simplifying, we get W = -F(ax^2 + bx^2).

Finally, we can evaluate the work done by substituting the limits of integration. Since we are moving from point 1x to 2x, the limits of integration are x = 1x to x = 2x.

Substituting the limits, we have W = -F(a(2x)^2 + b(2x)^2) - (-F(a(1x)^2 + b(1x)^2)).

Simplifying further, we get W = -F(4ax^2 + 4bx^2) + F(ax^2 + bx^2).

Combining like terms, we have W = -3F(ax^2 + bx^2).

Therefore, the work done by the variable force -2F(ax + bx) in moving the particle from point 1x to 2x along a straight line is -3F(ax^2 + bx^2).