A 3.0-m-long steel chain is stretched out along the top level of a horizontal scaffold at a construction site, in such a way that 2.0 m of the chain remains on the top level and 1.0 m hangs vertically, Fig. 7–27. At this point, the force on the hanging segment is sufficient to pull the entire chain over the edge. Once the chain is moving, the kinetic friction is so small that it can be neglected. How much work is performed on the chain by the force of gravity as the chain falls from the point where 2.0 m remains on the scaffold to the point where the entire chain has left the scaffold? (Assume that the chain has a linear weight density of 24 N/m.)

Verified step by step guidance

1. Determine the total weight of the chain using the linear weight density. The chain has a linear weight density of 24 N/m, so the total weight of the chain is given by multiplying the linear weight density by the total length of the chain: \( W_{\text{total}} = \lambda \cdot L \), where \( \lambda = 24 \ \text{N/m} \) and \( L = 3.0 \ \text{m} \).
2. Divide the chain into two segments: the portion initially on the scaffold (2.0 m) and the portion initially hanging vertically (1.0 m). Calculate the weight of each segment using \( W = \lambda \cdot L \).
3. Understand that as the chain falls, the center of mass of the chain moves downward. The work done by gravity is equal to the change in gravitational potential energy of the chain. The gravitational potential energy is given by \( U = m g h \), where \( h \) is the height of the center of mass of the chain.
4. Calculate the initial height of the center of mass of the chain. For the portion on the scaffold, the center of mass is at the midpoint of the 2.0 m segment, which is 1.0 m from the edge. For the hanging portion, the center of mass is at the midpoint of the 1.0 m segment, which is 0.5 m below the edge. Combine these to find the overall center of mass height initially.
5. Calculate the final height of the center of mass of the chain when it has completely fallen off the scaffold. At this point, the chain is fully vertical, and the center of mass is at the midpoint of the 3.0 m chain, which is 1.5 m below the edge. Subtract the final gravitational potential energy from the initial gravitational potential energy to find the work done by gravity.

Key Concepts

Work and Energy

Work is defined as the transfer of energy that occurs when a force is applied over a distance. In the context of this problem, the work done by gravity on the chain can be calculated by considering the change in potential energy as the chain falls. The formula for work done by a constant force is W = F × d, where F is the force and d is the distance moved in the direction of the force.

Linear Weight Density

Linear weight density is a measure of weight per unit length of an object, expressed in units such as N/m. In this scenario, the chain has a linear weight density of 24 N/m, which means that for every meter of the chain, it exerts a weight of 24 Newtons. This concept is crucial for calculating the total weight of the hanging portion of the chain, which directly influences the work done by gravity.

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field. For the chain, the potential energy decreases as it falls from the scaffold. The change in potential energy can be calculated using the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height change. This change in potential energy corresponds to the work done by gravity on the chain.