I am facing a problem in Physics.
Problem: What will be the work done by the frictional force over a polynomial curve if a body is sliding on this polynomial($a+bx+cx^2+dx^3+\ldots$) curve from rest from the height $h_1$ to height $h_2$ (where $h_1 > h_2$).
I tried to solve this as follows:
frictional force $F = k mg \cos\theta$, where $mg \cos\theta$ is normal force at that point. $k$ is coefficient of friction
Total work done=Line Integration over the polynomial(dot product of F and displacement).
But to go ahead from this point,i do not know.
Best Answer
I) The easy way to calculate the work $W_{\rm fric}$ done by friction (if one also knows initial and final speeds of the body, cf. DarenW's comment), is to use energy conservation
$$ W_{\rm fric}~=~ -\Delta E_{\rm kin} -\Delta E_{\rm pot}. $$
II) Else one would have to set up Newton's 2nd law along the curve, which is a second order vector-valued ODE, and solve it.