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?

Question

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?

Emily Abbate · Accepted Answer

#~~0.96#

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#

Parker Abbott · Answer

undefined The coefficient of kinetic friction is approximately 0.235.