An object with a mass of #6 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 2+cscx #. How much work would it take to move the object over #x in [pi/12, (5pi)/6]#, where x is in meters?
1 Answer
Explanation:
Work done by a variable force is defined as the area under the force vs. displacement curve. Therefore, it can be expressed as the integral of force:
#color(darkblue)(W=int_(x_i)^(x_f)F_xdx)# where
#x_i# is the object's initial position and#x_f# is the object's final position
Assuming dynamic equilibrium, where we are applying enough force to move the object but not enough to cause a net force and therefore not enough to cause an acceleration, our parallel forces can be summed as:
#sumF_x=F_a-f_k=0#
Therefore we have that
We also have a state of dynamic equilibrium between our perpendicular forces:
#sumF_y=n-F_g=0#
#=>n=mg#
We know that
#vecf_k=mu_kmg#
#=>color(darkblue)(W=int_(x_i)^(x_f)mu_kmgdx)#
We have the following information:
#|->"m"=6"kg"# #|->mu_k(x)=2+csc(x)# #|->x in[pi/12,(5pi)/6]# #|->g=9.81"m"//"s"^2#
Returning to our integration, know the
#color(darkblue)(W=mgint_(x_i)^(x_f)mu_kdx)#
Substituting in our known values:
#=>W=(6)(9.81)int_(pi/12)^((5pi)/6)2+csc(x)dx#
Evaluating:
#=>(6)(9.81)[2x+lnabs(csc(x)-cot(x))]_(pi/12)^((5pi)/6)#
#=>(58.86)[((5pi)/3+lnabs(2-(-sqrt3)))-(pi/6+lnabs(csc(pi/12)-cot(pi/12)))]#
#=>(58.86)[(3pi)/2+lnabs(2+sqrt3)-lnabs(csc(pi/12)-cot(pi/12))]#
#=>~~474.23#
Therefore, we have that the work done is
