Using a simple pulley and rope system, a crewman on an Arctic expedition is trying to lower a crate of mass \( m_{1}=6.04 \mathrm{~kg} \) to the bottom of a steep ravine of height \( H=27.2 \mathrm{~m} \). The crewman of mass \( m_{2}=55.6 \mathrm{~kg} \) is walking while holding the rope, being careful to lower the crate at a constant speed of \( v_{0}=1.50 \mathrm{~m} / \mathrm{s} \).

To solve this problem, we can use the concept of work and energy conservation. The work done by the crewman in lowering the crate is equal to the work done by the gravitational force on the crate. The work done by the crewman is given by: \[ W_{\text{crewman}} = F \cdot d \] where \( F \) is the force applied by the crewman and \( d \) is the distance over which the force is applied. The work done by the gravitational force on the crate is given by: \[ W_{\text{gravity}} = m_{1} \cdot g \cdot H \] where \( m_{1} \) is the mass of the crate, \( g \) is the acceleration due to gravity, and \( H \) is the height of the ravine. Since the crate is being lowered at a constant speed, the net force on the crate is zero. Therefore, the force applied by the crewman is equal to the gravitational force on the crate. We can set up the equation: \[ F = m_{1} \cdot g \] Now, we can calculate the force applied by the crewman and the work done by the crewman and the gravitational force on the crate. Calculate the value by following steps: - step0: Calculate: \(6.04\times 9.81\) - step1: Multiply the numbers: \(59.2524\) The force applied by the crewman is approximately \( 59.25 \, \text{N} \). Now, we can calculate the work done by the crewman and the gravitational force on the crate. Calculate the value by following steps: - step0: Calculate: \(59.25\times 27.2\) - step1: Multiply the numbers: \(1611.6\) The work done by the crewman is approximately \( 1611.6 \, \text{J} \). The work done by the gravitational force on the crate is: \[ W_{\text{gravity}} = m_{1} \cdot g \cdot H \] Substitute the values: \[ W_{\text{gravity}} = 6.04 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 27.2 \, \text{m} \] Now, we can calculate the work done by the gravitational force on the crate. Calculate the value by following steps: - step0: Calculate: \(6.04\times 9.81\times 27.2\) - step1: Multiply the terms: \(59.2524\times 27.2\) - step2: Multiply the numbers: \(1611.66528\) The work done by the gravitational force on the crate is approximately \( 1611.67 \, \text{J} \). Therefore, the work done by the crewman is equal to the work done by the gravitational force on the crate, which is approximately \( 1611.6 \, \text{J} \).