Question

An elevator cabin has a mass of \(358.1 \mathrm{~kg},\) and the combined mass of the people inside the cabin is \(169.2 \mathrm{~kg} .\) The cabin is pulled upward by a cable, with a constant acceleration of \(4.11 \mathrm{~m} / \mathrm{s}^{2}\). What is the tension in the cable?

Short answer

Answer: The tension in the cable is approximately \(7337.9\,\mathrm{N}\).

Step-by-step solution

1. Identify the given information and unknown

We are given: - Mass of the elevator cabin: \(m_{1}=358.1 \mathrm{~kg}\) - Mass of people inside the cabin: \(m_{2}=169.2 \mathrm{~kg}\) - Acceleration of the elevator: \(a= 4.11 \mathrm{~m/s^2}\) Unknown: - Tension in the cable: \(T\)

2. Calculate the total mass of the elevator system

To find the total mass of the elevator system (cabin and people inside), we add the masses of the cabin and the people: \(m_{total} = m_{1} + m_{2} = 358.1\,\mathrm{kg} + 169.2\,\mathrm{kg} = 527.3\,\mathrm{kg}\)

3. Calculate the weight of the elevator system

The weight of the elevator system (cabin and people inside) can be calculated using the equation: \(W = m_{total} \cdot g\) Where \(g = 9.81\,\mathrm{m/s^2}\) is the acceleration due to gravity. So, \(W = 527.3\,\mathrm{kg} \cdot 9.81\,\mathrm{m/s^2} = 5170.79\,\mathrm{N}\)

4. Apply Newton's second law

According to Newton's second law of motion, the net force acting on an object is equal to the product of its mass and acceleration: \(F_{net} = m_{total} \cdot a\) We can calculate the net force acting on the elevator system: \(F_{net} = 527.3\,\mathrm{kg} \cdot 4.11\,\mathrm{m/s^2} = 2167.113\,\mathrm{N}\)

5. Calculate tension in the cable

Now, we need to find the tension in the cable (\(T\)) supporting the elevator system. The upward force (tension) must be greater than the downward force (the weight) to make the elevator move upward with acceleration. Therefore, \(T - W = F_{net}\) Solving for \(T\): \(T = F_{net} + W = 2167.113\,\mathrm{N} + 5170.79\,\mathrm{N} = 7337.903\,\mathrm{N}\) So, the tension in the cable is approximately \(7337.9\,\mathrm{N}\).

Key concepts

Elevator Physics

Elevator physics deals with the forces and motions experienced by an elevator system when it is moving. Understanding the dynamics of an elevator requires examining several physical principles involving mass, acceleration, and force.
When an elevator moves upward or downward, it experiences changes in velocity and acceleration, influenced primarily by two key forces: tension and gravity.
As people board an elevator, the total mass of the system increases. This impacts the forces at play, as greater mass requires more tension in the supporting cables to achieve the same acceleration. The interaction between these forces is described by Newton's second law of motion, which helps determine the net force needed for the elevator to move safely and efficiently.

• Total mass is the sum of the elevator's own mass and the mass of its passengers.
• Acceleration in an elevator can be constant or variable, depending on its movement.

Understanding these principles helps in designing safe and efficient elevator systems capable of carrying passengers smoothly and securely.

Tension in Cable

The tension in the cable of an elevator is crucial for ensuring that the elevator moves as intended, especially when overcoming gravity to ascend. To understand how tension works in this context, we need to consider the forces acting on the elevator system.
Tension in the cable is the force that should support the total weight of the elevator system and provide it the additional force necessary to accelerate upwards. This tension is the result of the upward pulling force counteracting the downward gravitational force.
Mathematically, tension in the cable can be calculated using the formula:

• \[ T = F_{net} + W \]

where \( F_{net} \) is the net force generated by the elevator's acceleration, and \( W \) is the weight of the elevator system.
This formula arises from Newton's second law, where the net force required to move the elevator system upwards with a certain acceleration must be provided by the cable tension. Hence, the cable is always at work, maintaining the necessary balance for motion.

Net Force Calculation

Net force calculation is an essential concept in understanding elevator physics. Newton's second law of motion provides the framework for this calculation.
According to the law, the net force acting on an object is the product of the mass and its acceleration:

• \[ F_{net} = m_{total} \times a \]

The net force determines how much additional force is needed to move the elevator upwards at a specified acceleration, beyond simply counteracting gravity.
In the context of an elevator, this means the cable must exert enough tension not only to lift the total weight of the elevator system but also to accelerate it at the desired rate.

• Total mass (\( m_{total} \)) is a sum of the elevator's own mass and the mass of the passengers.
• Acceleration (\( a \)) is the rate at which the elevator is speeding up.

Understanding how to calculate and apply the net force is critical for designing systems that work reliably under various conditions, ensuring safety and efficiency as the elevator operates. These calculations typically involve considering multiple forces acting in different directions to find an equilibrium that supports smooth movement.