Question

An object, previously at rest, slides `5 m` down a ramp, with an incline of `pi/6`, and then slides horizontally on the floor for another `1 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)= 5m

Angle of inclination of the ramp `theta = pi/6`
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=5mgsintheta 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=mumgcosthetaxx5J`

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 6m along floor
`W_2=F_fxx3=3mumgJ`

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`
`:. 5mgsintheta=W_1+W_2`
`=>5mgsintheta=5mumgcostheta+1mumg`
`=>mu=(5sintheta)/(1+5costheta)=(5sin(pi/6))/(1+5cos(pi/6))`
`=(5xx1/2)/(1+5xxsqrt3/2)=2.5/(1+2.5xxsqrt3)=~~0.47`