Energy lost due to friction

The energy lost due to friction is the reduction in total energy of the object. The initial total energy is given by:

Ei=Ki+Ui=12mvi2+mghiEi=Ki+Ui=12mvi2+mghi

where KiKi is the initial kinetic energy, UiUi is the initial potential energy, vivi is the initial speed, hihi is the initial elevation above the base of the ramp, and gg is the acceleration of gravity.

The final total energy is given by:

Ef=Kf+Uf=12mvf2+mghfEf=Kf+Uf=12mvf2+mghf

where KfKf is the final kinetic energy, UfUf is the final potential energy, vfvf is the final speed, and hfhf is the final elevation above the base of the ramp.

In the case of a friction-free ramp, the initial total energy and the final total energy are equal. Their difference is zero. Energy is conserved. However, friction is a non-conservative force. The work done by n0n-conservative forces, WncWnc, is given by:

Wnc=ΔE=Ef−Ei=ΔK+ΔU=(Kf−Ki)+(Uf−Ui)Wnc=ΔE=Ef−Ei=ΔK+ΔU=(Kf−Ki)+(Uf−Ui)

In the case of a ramp inclined at angle θθ, the change in elevation is given by:

Δh=hf−hi=ssinθΔh=hf−hi=ssin⁡θ

where ss is the distance that the object slides up the ramp. In this case, the change in potential energy is given by

ΔU=mgssinθΔU=mgssin⁡θ.