An object, previously at rest, slides #15 m# down a ramp, with an incline of #pi/3 #, and then slides horizontally on the floor for another #6 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

1 Answer
Apr 9, 2016

Answer:

#~~0.96#

Explanation:

Here length of the ramp (l)= 15m

Angle of inclination of the ramp #theta = pi/3#
Height of the object from horizontal floor,#h=lsintheta#
If mass ofthe body is #m kg#
The initial gravitational potential energy of the body #PE=mgh=mglsintheta=15mgsintheta 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 kinetc friction
work done against frictional force when sliding down the ramp #W_1=F_rxxl=mumgcosthetaxx15J#

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_fxx6=6mumgJ#
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#
#:. 15mgsintheta=W_1+W_2#
#=>15mgsintheta=15mumgcostheta+6mumg#
#=>mu=(15sintheta)/(6+15costheta)=(5sin(pi/3))/(2+5cos(pi/3))#
#=(5xxsqrt3/2)/(2+5xx1/2)=(2.5xxsqrt3)/4.5=5/9xxsqrt3=~~0.96#