A particle moving along the x -axis is subjected to a variable force F = c x^n , where c and n are constants. It is observed that the work done by this force in displacing the particle from the origin ( x = 0 ) to a position x₀ is directly proportional to x₀^3 . How does the force F depend on the position x ?

Step-by-step solution

The work done by a variable force is calculated by integrating the force over the displacement. W = ₀^ x₀ F , dx Substitute the given force expression F = c x^n : W = ₀^ x₀ c x^n , dx Applying the power rule for integration: W = c [ x^ n+1 n+1 ]₀^ x₀ = c n+1 x₀^ n+1 This shows that the work done W is proportional to x₀^ n+1 . The problem states that W x₀^3 . By comparing the exponents, we get: n + 1 = 3 n = 2 Substituting n = 2 back into the force equation gives F = c x^2 . Therefore, the force is directly proporti