If h is the height the object moves up the plane and l is the slant distance moved along the plane and d is the run.
h/l=sintheta
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Force of friction F_f=muN=mumg cos theta
Where mu is the coefficient of dynamic friction
m is the mass
g is gravity
theta is the angle above horizontal
Work done against friction:
"work done" = "force due to friction" xx "distance moved "l
= (muxxmgcostheta) xx l
When the object comes to stop, part of its Kinetic energy E got converted into its potential energy and rest of it has been used to overcome kinetic friction.
mgh=E- mumgcos thetaxxl
mglsintheta=E- mumgcos thetaxxl, solving for l
=>l =E/(mgsintheta+mumgcostheta)
Again
"work done" = "force due to friction" xx "distance moved "l
= (muxxmgcostheta) xx l
=(muEmgcostheta)/(mgsintheta+mumgcostheta)=
(muEcostheta)/(sintheta+mucostheta)