Calculate the instantaneous power delivered to a mass by time variable force
Problem:(This problem is from above Youtube video)
A crate of mass 4 kg is pulled from rest with a force whose magnitude is given by F = 5t^2. If the crate accelerates with a constant acceleration of 3 m//s^2, determine the instantaneous power delivered to the crate by the force at t = 5 s. Assume that the surface is smooth. ======= First of all, the problem itself violates Newton's 2nd law, F = ma. Since mass m is a constant, if force is changed by time than the crate cannot keep a constant acceleration. If set the initial velocity and initial acceleration equal 0 this problem is still solvable. ======= Solution: vec(F)=mvec(a) => vec(a) = vec(F)/m = (5t^2)/(4 kg) = 5/4t^2 vec(v)= = int vec(a) dt = int(5/4t^2)dt = v_0 + 5/12t^3 Since we set v_0 = 0 vec(v) = 5/12t^3 Power: P = Fv P = 5t^2 × 5/12t^3 = 25/12t^5 At t = 5 s P = 25/12(5)^5 = 6.5 × 10^3 W