91Ó°ÊÓ

When lifting a pencil off the top of a desk, one is exerting an upward force against the force of gravity to move the pencil. In this case, the force exerted is the force due to gravity or weight, which can be calculated as: \(F_{gravity} = m \times g\) where, \(F_{gravity}\) is the gravitational force, \(m\) is the mass of the pencil, and \(g\) is the acceleration due to gravity (\(9.81 m/s^2\)). Since the pencil is moved against the force of gravity, work is being performed. The work done against gravity when lifting the pencil at a height h can be calculated using the formula: \(W = F_{gravity} \times h\) where \(W\) is the work done.

In this case, one compresses a spring to half its normal length, which requires an application of a force to change the spring's position. The force exerted in this case is the spring force, which follows Hooke's Law and can be calculated as: \(F_{spring} = -k \times x\) where \(F_{spring}\) is the spring force, \(k\) is the spring constant, and \(x\) is the displacement from the spring's equilibrium position. Here, the negative sign indicates that the spring force is always opposing the displacement direction. Since the spring is compressed, work is being performed in this case as well. The work done in compressing the spring can be calculated using the formula: \(W = \dfrac{1}{2} k \times x^2 \) where \(W\) is the work done. In summary, work is being performed in both cases (a) and (b). In the first case, work is done against gravity to lift a pencil off the desk, while in the second case, work is done to compress a spring to half its normal length.