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Math Help - Work, energy, power?

  1. #1
    Super Member fardeen_gen's Avatar
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    Work, energy, power?

    A body of mass m slides down a plane inclined at an angle \alpha. The coefficient of friction is \mu. Find rate at which kinetic plus gravitational potential energy is dissipated at any time t.
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  2. #2
    Super Member Rebesques's Avatar
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    Motion exists along the surface of the plane only (call this direction horizontal). Analyse the weight force B=mg in two components: the vertical and horizontal forces, N=mg\cos(a) and F=mg\sin(a) respectively. Along the horizontal direction, two forces are applied: F and the friction T=\mu m g \cos(a). Call v(t) the velocity at time t. By Newton's law, F-T=mv'(t), and solve to get v(t)=g(\sin(a)-\mu \cos(a))t (assume the body was resting at t=0).
    If h(t) is the height and S(t)=\int_0^t v(s)ds the length traveled at time t, then a simple argument shows h(t)=S(t)\sin(a). Now, the total energy is E(t)=mgh(t)+1/2mv^2(t)=mgS(t)\sin(a)+1/2mv^2(t). Differentiate to get E'(t)=mgv(t)\sin(a)+mv(t)v'(t)...


    (Sorry for being so analytic. I was so bored today )
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