Question

An object, previously at rest, slides `9 m` down a ramp, with an incline of `(pi)/3`, and then slides horizontally on the floor for another `2 m`. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Answer 1

Here length of the ramp (l)= 9m

Angle of inclination of the ramp `theta = pi/3`
Height of the object from horizontal floor,`h=lsintheta`
If mass of the body is `m kg`
The initial gravitational potential energy of the body `PE=mgh=mglsintheta=9mgsintheta J`

The normal reaction acting on the body when it is sliding down the ramp is `N_r=mgcostheta`
and the corresponding frictional force `F_r=muN_r=mumgcostheta`
where `mu=`coefficient of kinetic friction
work done against frictional force when sliding down the ramp `W_1=F_rxxl=mumgcosthetaxx9J`

when the body slides on horizontal force,then normal reaction `N_f=mg` and corresponding frictional force `F_f=muN_f=mumg`
work done against frictional force when sliding 2m along floor
`W_2=F_fxx2=2mumgJ`

Now applying conservation of mechanical energy we can write

The initial KE being zero
Initial PE = total work done against frictional force `=W_1+W_2`
`:. 9mgsintheta=W_1+W_2`
`=>9mgsintheta=9mumgcostheta+2mumg`
`=>mu=(9sintheta)/(2+9costheta)=(9sin(pi/3))/(2+9cos(pi/3))`
`=(9xxsqrt3/2)/(2+9xx1/2)=(4.5xxsqrt3)/6.5=9/13xxsqrt3> 1` not feasible