If a #7/2 kg# object moving at #9/4 m/s# slows to a halt after moving #3/8 m#, what is the coefficient of kinetic friction of the surface that the object was moving over?

Question

Adrian Ayala · Accepted Answer

#mu_k = 0.306#

We're asked to find the coefficient of kinetic friction #mu_k# of an object sliding on a surface, with some given information.

While sliding, the vertical forces (gravitational and normal) cancel out, so we'll look at only horizontal motion to find the friction.

We can use kinematics to find the magnitude of the acceleration, given it went from #9/4# #"m/s"# to rest in #3/8# #"m"#:

#(v_x)^2 = (v_(0x))^2 + 2a_x(Deltax)#

We know:

#v_x = 0# (comes to rest)

#v_(0x) = 2.25# #"m/s"#

#Deltax = 0.375# #"m"#

#0 = (2.25color(white)(l)"m/s")^2 + 2a_x(0.375color(white)(l)"m")#

#color(red)(a_x = 3# #color(red)("m/s"^2#

Now that we know the acceleration, we can use Newton's second law to find the magnitude of the friction force:

#sumF_x = ma_x = (7/2color(white)(l)"kg")(color(red)(3color(white)(l)"m/s"^2)) = 10.5# #"N"# #= f_k#

The coefficient of kinetic friction #mu_k# is given by

#f_k = mu_kn#

The normal force #n# is

#n = mg = (7/2color(white)(l)"kg")(9.81color(white)(l)"m/s"^2) = 34.3# #"N"#

The coefficient of kinetic friction is thus

#mu_k = (f_k)/n = (10.5cancel("N"))/(34.3cancel("N")) = color(blue)(0.306#

Elena Albarran · Answer

undefined To find the coefficient of kinetic friction, we can use the formula:

$$ \text{Coefficient of kinetic friction} = \frac{\text{Force of friction}}{\text{Normal force}} $$

The force of friction can be calculated using Newton's second law:

$$ \text{Force of friction} = \text{Mass} \times \text{Acceleration} $$

The acceleration can be calculated using the equation of motion:

$$ v^2 = u^2 + 2as $$

Where:
$ v $ = final velocity (0 m/s, since the object comes to a halt)
$ u $ = initial velocity ($ \frac{9}{4} $ m/s)
$ a $ = acceleration
$ s $ = displacement ($ \frac{3}{8} $ m)

Solve for acceleration ($ a $), then use it to find the force of friction. Finally, divide the force of friction by the normal force to get the coefficient of kinetic friction.