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`