Question

4.41 A bosun's chair is a device used by a boatswain to lift himself to the top of the mainsail of a ship. A simplified device consists of a chair, a rope of negligible mass, and a frictionless pulley attached to the top of the mainsail. The rope goes over the pulley, with one end attached to the chair, and the boatswain pulls on the other end, lifting himself upward. The chair and boatswain have a total mass \(M=90.0 \mathrm{~kg}\). a) If the boatswain is pulling himself up at a constant speed, with what magnitude of force must he pull on the rope? b) If, instead, the boatswain moves in a jerky fashion, accelerating upward with a maximum acceleration of magnitude \(a=2.00 \mathrm{~m} / \mathrm{s}^{2},\) with what maximum magnitude of force must he pull on the rope?

Short answer

The force the boatswain must apply when lifting himself at a constant speed is 882.90 N. When lifting himself with a maximum acceleration of 2.00 m/s², he must apply a force of 1062.90 N.

Step-by-step solution

Identify the forces acting on the boatswain and the chair

In both situations, the gravitational force acts downward on the boatswain and the chair, and the tension force in the rope acts upward. The gravitational force is given by \(F_g = Mg\), where \(M = 90.0 \mathrm{~kg}\) (mass of boatswain and chair) and \(g = 9.81 \mathrm{~m}/\mathrm{s}^{2}\) (acceleration due to gravity).

Calculate the force needed to lift at constant speed

When the boatswain is lifting himself at a constant speed, there is no acceleration, which means that the net force on the system must be zero. In other words, the gravitational force and the tension force in the rope must be equal in magnitude. Therefore, the force the boatswain must apply on the rope is equal to the gravitational force: \(F_t = F_g = Mg = (90.0 \mathrm{~kg})(9.81 \mathrm{~m}/\mathrm{s}^{2}) = 882.90 \newton\).

Calculate the force needed to lift with maximum acceleration

In the second scenario, the boatswain accelerates upward with the maximum acceleration \(a = 2.00 \mathrm{~m}/\mathrm{s}^{2}\). According to Newton's second law, the net force acting on the system is given by \(F_{net} = Ma\). The net force is the difference between the tension force (\(F_t\)) and the gravitational force (\(F_g\)): \(F_{net} = F_t - F_g\). By applying Newton's second law of motion, we have \(F_t - F_g = Ma\). Now we can solve for the maximum force: \(F_t = Ma + F_g = M(a + g)\). \(F_t = (90.0 \mathrm{~kg})(2.00 \mathrm{~m}/\mathrm{s}^{2} + 9.81 \mathrm{~m}/\mathrm{s}^{2}) = (90.0 \mathrm{~kg})(11.81 \mathrm{~m}/\mathrm{s}^{2}) = 1062.90 \newton\). #Step 4: Summarize the results# a) When the boatswain lifts himself at a constant speed, he must apply a force of \(882.90 \newton\) on the rope. b) When the boatswain lifts himself with a maximum acceleration of \(2.00 \mathrm{~m}/\mathrm{s}^{2}\), he must apply a force of \(1062.90 \newton\) on the rope.

Key concepts

Newton's second law of motion

Newton's second law of motion is a fundamental principle that describes the relationship between an object's mass, the net force acting on it, and its acceleration. It is often summarized by the equation \(F_{net} = ma\), where \(F_{net}\) represents the net force on the object, \(m\) is the object's mass, and \(a\) is the acceleration.

In the context of the bosun's chair physics problem, when the boatswain pulls himself upward with a constant speed, the net force is zero because his velocity is constant, and he is not accelerating. This means the upward tension force is perfectly balanced by the downward gravitational force. However, when he accelerates upward, the net force is no longer zero. According to Newton's second law, the tension must be greater than the gravitational force to provide this upward acceleration. Hence, the maximum force he needs to apply is calculated by adding the product of mass and acceleration (\(Ma\)) to the gravitational force, which is essentially applying Newton's second law to find the required tension in the rope.

Gravitational force

The gravitational force, also known as weight, is the force with which a mass is attracted towards the center of a celestial body, such as the Earth. It is calculated using the formula \(F_g = mg\), where \(m\) is the mass of the object and \(g\) is the acceleration due to gravity, which averages about \(9.81 \text{m/s}^2\) on Earth's surface.

For our bosun's chair exercise, the gravitational force is what the boatswain must overcome to ascend. It acts downward on both the boatswain and the chair with a force directly proportional to their combined mass. At constant speed, the tension force must match this gravitational force exactly, leading to the solution for the first part where the needed force equals \(882.90 ewton\). This force is equal to the gravitational pull on the 90.0 kg mass of the boatswain plus the chair.

Tension force in physics

Tension force is a pulling force transmitted through a string, cable, or rope when it's pulled tight by forces acting from opposite ends. The force is directed along the length of the wire and pulls equally on the objects on the opposite ends of the wire.

In our bosun's chair scenario, the rope exerts tension that counteracts the gravitational force to either maintain a steady position or allow the boatswain to ascend. When he moves up at a constant speed, the tension in the rope equals the weight of the boatswain and chair combined. When accelerating upwards, the tension must be greater to not only balance the weight but also provide the extra force required for the acceleration, resulting in a larger tension force in the rope, which, as the calculations show, is \(1062.90 ewton\). This tension force is critical for understanding how systems like pulleys operate and how forces are distributed within such systems.