An object, previously at rest, slides #4 m# down a ramp, with an incline of #pi/4 #, and then slides horizontally on the floor for another #15 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 #4  m# down a ramp, with an incline of #pi/4 #, and then slides horizontally on the floor for another #15 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Emily Abbate · Accepted Answer

#u_k~=0,158# #"Total Energy of object:"# #E_p=m*g*4*sqrt2/2" "E_p=m*g*2sqrt2# #"Work doing by the Friction Force on Ramp"# #W_f=u_k*m*g*4*sqrt2/2" "W_f=u_k*m*g*2sqrt 2# #"Total Energy on floor"# #Delta E=m*g*2sqrt2-u_k*m*g*2sqrt2# #color(red)(W_f=(u_k*m*g)*15)# #color(red)"work doing by The Friction Force on Floor"# #Delta E=cancel(m*g)*2sqrt2-u_k*cancel(m*g)*2sqrt2=color(red)(u_k*cancel(m*g)*15# #2sqrt2-u_k*2sqrt2=u_k*15# #2sqrt2=15*u_k+2sqrt2*u_k# #2,82=15*u_k+2,82*u_k# #2,82=17,82*u_k# #u_k=(2,82)/(17,82)# #u_k~=0,158#

Nathan Ashcroft · Answer

undefined To find the coefficient of kinetic friction, we first calculate the total distance traveled horizontally using the ramp and the horizontal distance on the floor. Then, we use the work-energy principle to relate the work done by friction to the change in kinetic energy of the object. Solving for the coefficient of kinetic friction gives us:

μ = (mgd - ΔKE) / (mgh)

where:
μ = coefficient of kinetic friction
m = mass of the object
g = acceleration due to gravity
d = horizontal distance on the ramp
h = vertical height of the ramp
ΔKE = change in kinetic energy

Given:
d = 4 m
h = 4 m (since the incline of π/4 radians means a 45-degree incline)
ΔKE = 0 (the object starts and ends at rest)

Using trigonometry, we find the horizontal distance traveled on the ramp:
d = h * tan(π/4) = 4 * tan(π/4) = 4 m

Total horizontal distance traveled = distance on ramp + distance on floor
Total horizontal distance = 4 m + 15 m = 19 m

Now, we can find the coefficient of kinetic friction:
μ = (mgd - ΔKE) / (mgh)
   = (mg(d + 15) - 0) / (mgh)
   = (mg(4 + 15)) / (mg * 4)
   = (19g) / (4g)
   = 19 / 4

Therefore, the coefficient of kinetic friction is 19/4 or 4.75.