An object with a mass of #1 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 2e^xx+3 #. How much work would it take to move the object over #x in [1, 4]#, where x is in meters?
1 Answer
Answer:
Explanation:
We're asked to find the work necessary to push an object on a certain position interval with a varying coefficient of kinetic friction.
The equation for work with a varying force is
#W = int_(x_1)^(x_2) F_xcolor(white)(l)dx#
where

#x_1# and#x_2# are the initial and final positions 
#F_x# is the force, which will be equal to the friction force acting (but in the opposite direction):
#F_x = mu_kn = mu_kmg#
So
#W = int_(x_1)^(x_2) mu_kmgcolor(white)(l)dx#
We know:

#x_1 = 1# #"m"# 
#x_2 = 4# #"m"# 
#mu_k = 2e^x  x + 3# 
#m = 1# #"kg"# 
#g = 9.81# #"m/s"^2#
Plugging these in:
#W = int_(1color(white)(l)"m")^(4color(white)(l)"m")(1color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)(2e^xx+3)color(white)(l)dx = color(red)(ulbar(stackrel(" ")(" "1033color(white)(l)"J"" "))#