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?

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?

Adrian Ayala · Accepted Answer

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#

Elena Albarran · Answer

undefined To find the kinetic friction coefficient (μ) on the horizontal floor, use the following equation:

$$ \mu = \tan(\theta) $$

where $ \theta $ is the angle of the incline (π/6). Therefore,

$$ \mu = \tan(\pi/6) $$