How much work is done on the cart by the rope?

Question

A cart loaded with bricks has a total mass of 12.3 kg and is pulled at constant speed by a rope. The rope is inclined at 24.5 degrees above the horizontal and the cart moves 12.5 m on a horizontal surface. The coefficient of kinetic friction between ground and
cart is 0.503.

The acceleration of gravity is 9.8 m/ s^2 :

How much work is done on the cart by the rope? Answer in units of J.

Answer 1

Construct a free body diagram.

Let's solve without numbers first
N=Normal Force
T=Rope Tension
μ=kinectic coeff.
θ=angle with horizontal
m=mass
g=gravity acceleration
s=distance

x Forces: μN-Tcos(θ)=0 (I)
y Forces: Tsin(θ)+N-mg (II)

First, solve for N by eliminating T by using (I) into (II)
we get
N=mg/(μtan(θ)+1)

The force that does the work (recall F.s) is in the x direction

so, it is Tcos(θ). but this is the same as μN
s0 the work will be
W=mgμs/(μtan(θ)+1)

plugging all the values in we get 616.6 J (it's all in MKS so units work)