Identify the force present and explain whether work is being performed in the following cases: (a) You lift a pencil off the top of a desk. \((\mathbf{b})\) A spring is compressed to half its normal length.

In this case, we have two main forces acting on the pencil: gravitational force (mg) and the applied force (F) used to lift the pencil. The gravitational force acts downward, while the applied force acts upward. Since the pencil is being lifted upward, the applied force is greater than the gravitational force. To determine whether work is being done, let's consider the work formula: \[ W = Fd\cos{\theta} \] where W is work, F is the applied force, d is the distance the pencil travels upwards, and θ is the angle between the applied force and the distance. In this case, the applied force and the distance moved by the pencil are in the same direction, so the angle θ is 0 degrees, and cos(0) = 1. Therefore, the work done in this case is: \[ W = Fd \] Since the pencil is being lifted (Fd > 0), work is performed on the pencil in this case.

In this scenario, we have a spring compressed to half of its normal length. The main force involved is the spring force, which acts to restore the spring to its uncompressed state. The spring force is given by Hooke's Law: \[ Fs = -kx \] where Fs is the spring force, k is the spring constant, and x is the compression of the spring. Again, to determine whether work is being done, let's consider the work formula: \[ W = Fd\cos{\theta} \] In this case, the force (Fs) and the distance (x) are in the same direction as the spring is compressed horizontally, so the angle θ is 0 degrees and cos(0) = 1. Therefore, the work done in this case is: \[ W = Fsx \] Since the spring is being compressed (Fsx > 0), work is being performed on the spring in this case.