Work-Energy Theorem EX 9

(a) The diagram below shows the forces acting on the mass and a free body diagram. The displacement of the mass is $\mathrm{\Delta }x=x$ because we chose the initial position of the mass as the origin of the coordinate system ($x_i=0$ and $x_f=x$).

1. The work done by the normal force is zero because it is perpendicular to the displacement or because the x-component of the normal force is zero.
   
   
2. The gravitational force is a constant force. The x-component of the gravitational force is ${F_G}_x=F_{G}cos(-53^{\circ })=(10)(0.6)=6N$ and so we may use $W_{F_G}={F_G}_x\mathrm{\Delta }x=6xJ$. We cannot compute its value because we don't know the value of x.
   
   
3. Before we compute the work done on the mass by friction we have to first determine the magnitude of the force of friction. Applying the Second Law to the y-components of forces we find that the normal force is 8 N and that, given the coefficient of friction is ${\mu }_k=0.1$, the force of friction is $F_f=(0.1)(8)=0.8N$. The x-component of the frictional force is ${F_f}_x=F_f\cos (180^{\circ })=-0.8N$. From there we find the work done by friction $W_{F_f}={F_f}_x\mathrm{\Delta }x=-0.8xJ$. Again, we cannot compute the value of this force because we don't know the value of x.
   
   
4. The force exerted on the mass by the spring is not constant. We have to use a graph to find it. As the block slides down the x-component of the spring force is ${F_{sp}}_x=F_{sp}cos(180^{\circ })=-F_{sp}$. Note that on the vertical axis we are plotting the x-component of the spring force which is negative (the graph is below the axis).
   
   

The area under the graph is $(\frac{1}{2}x)\times (-kx)=-\frac{1}{2}k{x}^2=-\frac{1}{2}250x^2=-125x^{2}J$. Therefore $W_{F_{sp}}=-125x^{2}J$.

(b) The net work is zero because the mass starts from rest and stops when it slides down the distance $x$. The change of the kinetic energy is zero.

(c) Knowing that $W_{net}=0=W_{F_G}+W_{F_f}+W_{F_{sp}}$ we can compute $x$. By substituting the expressions found above into this equation we write:

$0=6x-0.8x-125x^2$

Solving for $x$ we find

$x=0.0416=4.16cm$.