Question

(b) Let F(r) be a force vector so that the line integral F(r)r is the work W done by F(r) along the curve C with parametric representation r(r) which is the displacement vector. Let m be the mass of the object. Prove that the work done on the object within te[to4 satisfies the equation W- cF(r).dr-îm v(r 2Г which states the work energy theorem: work done on an object equals to the change of kinetic energy (Hint I: Newton's second law) (Hint 2: (f).f())-2).o)) to

Answer 1

Step 1: From Newton's second law of motion, Force = mass times accelaration. i.e. in vector notation F = ma = m d^2 r (t)/dt^2 where r (t) is the displacement vector. Step 2: The work done W by the force F (r) is given by W = Integral_3 F (r) middot dr = Integral^t_1_t_0 F (r (t)) middot dr (t)/dt dt = Integral^t_1_t_0 m d^2 r (t)/dt^2 middot (dr (t)/dt dt = m Integral^t_1_t_0 d/dt (dr (t)/dt) middot dr (t)/dt dt = m Integral^t_1_t_0 dv (t)/dt middot V (t) dt [Where V (t) = dr (t)/dt = velocity vector] Therefore w = m Integral^t_1_t_0 V (t) middot dv (t) = m/w Integral^t_1_t_0 2 (V (t) middot dv (t)) = m/2 Integral^t_1_t_0 [V (t) middot dv (t) + dv (t) middot V (t)] = m/2 Integral^t_1_t_0 d (V (t) middot V (t)) [Because (f (t) middot f' (t)) = 2f (t) middot f' (t)]