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 7336.29 N.

Step-by-step solution

Identify the known variables

We are given the following information: - Mass of the elevator cabin (m1) = \(358.1 \mathrm{~kg}\) - Combined mass of the people (m2) = \(169.2 \mathrm{~kg}\) - Acceleration (a) = \(4.11 \mathrm{~m/s^2}\)

Calculate the total mass

Add the mass of the elevator and the combined mass of the people to find the total mass. Total Mass (M) = m1 + m2 M = 358.1 kg + 169.2 kg M = \(527.3 \mathrm{~kg}\)

Apply Newton's second law of motion

To find the tension (T) in the cable, we use Newton's second law (F = ma) where F is the net force, m is the total mass, and a is the acceleration. The tension T acts upward, and the gravitational force (W) acts downward: W = Mg where W is the gravitational force, M is the total mass, and g is the acceleration due to gravity (\(9.81 \mathrm{~m/s^2}\)).

Calculate the gravitational force

Using the equation for gravitational force, we can find W: W = Mg W = 527.3 kg * 9.81 m/s² W = \(5169.10 \mathrm{~N}\) (rounded to two decimal places)

Calculate the net force acting on the elevator

Now, we will determine the net force (F) acting on the elevator using Newton's second law (F = ma): F = Ma F = 527.3 kg * 4.11 m/s² F = \(2167.19 \mathrm{~N}\) (rounded to two decimal places)

Calculate the tension in the cable

Finally, we can find the tension (T) in the cable by adding the gravitational force (W) and the net force (F): T = W + F T = 5169.10 N + 2167.19 N T = \(7336.29 \mathrm{~N}\) (rounded to two decimal places) The tension in the cable is \(7336.29 \mathrm{~N}\).

Key concepts

Newton's second law of motion

Newton's second law of motion is a fundamental principle that describes the relationship between the motion of an object and the forces acting upon it. In the simplest terms, this law states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, we can express this as the equation:

\( F = ma \)

where \( F \) represents the net force applied to the object, \( m \) is the mass of the object, and \( a \) is the acceleration of the object. When solving physics problems involving motion, like in the case of the elevator exercise, this law is crucial.

To apply Newton's second law to determine the tension in an elevator cable, we start by identifying all the forces acting on the elevator system. These include the tension in the cable pulling upwards and the gravitational force pulling downwards. The tension in the cable must not only support the weight of the system but also provide the additional force necessary to accelerate the elevator. This is where our understanding of Newton's second law becomes especially important, as it guides us in calculating the exact force needed to achieve the desired acceleration.

Gravitational force

Gravitational force is a naturally occurring phenomenon that tugs at all masses in the universe, curiously drawing them toward one another. Perhaps the most familiar manifestation of this force is the attraction that Earth exercises on objects, giving them weight and ensuring they don't float away into space. For calculations, the gravitational force \( W \) that an object experiences due to Earth's gravity can be determined using the equation:

\( W = Mg \)

Here, \( M \) is the object's mass and \( g \) is the acceleration due to gravity, which on Earth's surface is approximately \( 9.81 \ m/s^2 \). Understanding gravitational force is essential when working with elevator physics or any scenario where objects are in a state of free fall or supported against gravity. In our elevator problem, the gravitational force represents a constant downward pull acting on both the elevator cabin and the people inside. Calculating this force is a critical step in determining the total force that the cable must exert to lift the elevator, which, as we'll explain in the next section, requires a combination of supporting the weight and providing additional force for acceleration.

Elevator physics

When discussing elevator physics, we're delving into the principles that govern the movement of an elevator system, which includes the cabin, any passengers or contents, and the counteracting forces. One integral factor in elevator scenarios is the tension in the cable, which must overcome the gravitational force acting on the elevator to either keep it stationary or to move it with a certain acceleration.

The tension in the cable isn't simply equal to the weight of the elevator system; it also has to account for the net force required to accelerate the system. When an elevator accelerates upward or downward, the cable's tension will vary in response. In the case of our elevator problem, the elevator is in an accelerated upward motion, so the tension must be greater than the gravitational force alone. By using Newton's second law of motion, we can understand that the cable must exert a force equal to the total mass of the system multiplied by the acceleration plus the force needed to counteract the gravitational pull. Only by addressing both these aspects can we correctly compute the tension in the cable, which ensures the elevator operates safely and effectively.