A particle moves along the x -axis under the action of a variable force given by F = a x^2 , where F is in newtons, x is in meters, and a is an unknown constant. If the work done by this force in moving the particle from x = 1 m to x = 2 m is 10 J , what is the value of the constant a in SI units?

Step-by-step solution

The work done by a variable one-dimensional force is given by the definite integral of the force with respect to position. W = _ x₁ ^ x₂ F , dx Substitute the given force function and limits: W = ₁² a x^2 , dx Integrate using the power rule ( x⁻² dx = -x⁻¹ ): W = a [ - 1 x ]₁² Evaluate the limits: W = a ( - 1 2 - (- 1 1 ) ) = a ( 1 - 1 2 ) = a 2 Equating this to the given work done of 10 J : a 2 = 10 a = 20 The value of the constant a is 20 in SI units. Answer: 20