Answer:
a)
There are two forces acting on the pencil when you lift the pencil off the top of the desk:
1) The applied force: this is the force applied by your hand. The direction of this force is upward. The work done by a force is given by
[tex]W=Fd cos \theta[/tex]
where
F is the magnitude of the force
d is the displacement of the object
[tex]\theta[/tex] is the angle between the direction of the force and the displacement
In this case, the applied force is upward, and the displacement is upward as well, so [tex]\theta=0^{\circ} \rightarrow cos \theta = +1[/tex], therefore the work done by this force is positive.
2) The gravitational force: this is the weight of the pencil, [tex]mg[/tex] (where m is the mass of the pencil and g is the acceleration due to gravity). The direction of this force is downward, while the displacement of the pencil is upward; therefore,
[tex]\theta=180^{\circ} \rightarrow cos \theta = -1[/tex]
which means that the force done by gravity is negative.
b)
Here we have a spring which is compressed to half its normal length.
Again, we have two forces acting on the spring (we ignore gravity for the purpose of this exercise):
1) The applied force --> this is the force that causes the spring to compress. Again, the work done by a force is
[tex]W=Fd cos \theta[/tex]
where
F is the magnitude of the force
d is the displacement of the object
[tex]\theta[/tex] is the angle between the direction of the force and the displacement
In this case, the applied force is in the same direction as the displament (the spring is compressed in the same direction as the external force), so [tex]\theta=0^{\circ} \rightarrow cos \theta = +1[/tex], therefore the work done by this force is positive.
2) The restoring force --> when a spring is compressed, the spring exerts a restoring force that tries to put the spring back into its equilibrium position. Therefore, the direction of the restoring force is opposite to the direction of the applied force, and so it is also opposite to the displacement of the spring.
So we have,
[tex]\theta=180^{\circ} \rightarrow cos \theta = -1[/tex]
which means that the force done by the restoring force is negative.
