Question

An elevator and its load have a combined mass of \(1600 \mathrm{~kg}\). Find the tension in the supporting cable when the elevator, originally moving downward at \(12.0 \mathrm{~m} / \mathrm{s}\), is brought to rest with constant acceleration in a distance of \(42.0 \mathrm{~m}\).

Short answer

The tension in the supporting cable when the elevator is brought to rest is \(11103.6 \, N\).

Step-by-step solution

Calculating the Deceleration of the Elevator

We first need to consider that this problem involves deceleration. We can use the equation of motion \( v^2 = u^2 + 2a s \), where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration (or deceleration in this case), and \(s\) is the distance covered. Given that final velocity \(v = 0 \, m/s\), initial velocity \(u = 12.0 \, m/s\), and the distance s = 42.0 m, we can solve for \(a\).

Determining the Actual Acceleration

On rearranging the equation, we get: \(a = (v^2 - u^2) / (2s)\). Substituting the values given, we get: \(a = (0 - (12.0)^2) / (2*42.0) = -3.429 \, m/s^2\). Notice that \(a\) is negative, which indicates that it's deceleration.

Calculating the Tension in the Cable

Next, we calculate the tension in the cable. Using Newton's second law \(F = m*a\), where \(F\) is the net force, \(m\) is the mass, and \(a\) is the acceleration, and given that the weight \(W = m*g = 1600*9.8 = 15680 \, N\) acts downward, and the tension \(T\) acts upward. Since the elevator is decelerating, which implies that the upward force is greater, we can say \(F = T - W\), hence \(T = m*a + W\). Now we substitute the values to solve for \(T\): \(T = 1600*(-3.429) + 15680 = 11103.6 \, N\).

Interpreting the Result

Hence, the tension in the supporting cable when the elevator is brought to rest is \(11103.6 \, N\). It is less than the weight of the elevator when at rest because the elevator is moving downward.

Key concepts

Deceleration Calculation

Understanding how to calculate deceleration is essential when examining scenarios where an object is slowing down, such as an elevator coming to a stop. Deceleration is simply negative acceleration, which means the object is reducing its speed over time. In physics problems, we often use kinematic equations to find deceleration.

One such equation is: \[v^2 = u^2 + 2a s\], where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the deceleration, and \(s\) is the distance over which the deceleration happens. If the object comes to a complete stop, then \(v = 0\) m/s. By rearranging the equation, we can solve for \(a\): \[a = \frac{v^2 - u^2}{2s}\]. It's important to note that the deceleration will be a negative value since it's acting in the opposite direction of the motion. This value is crucial for solving further parts of the problem, like determining forces involved, such as the tension in the cable of the elevator.

Newton's Second Law

Newton's second law of motion is fundamental when dealing with forces and motion. It states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration: \(F = m*a\). This law allows us to connect the dots between an object's motion and the forces acting upon it.

In the case of an elevator, this means that the total of all forces acting on it will determine its acceleration (or deceleration, as it's the same thing but in the opposite direction). For an object moving downwards and slowing down, such as our elevator, the net force is upward. This net force is made up of the tension in the supporting cable minus the weight of the elevator, which is always directed downward due to gravity (\(W = m*g\), where \(g\) is the gravitational acceleration). Therefore, the net force will be the tension in the cable (upward force) minus the object's weight (downward force).

Tension in a Cable

Tension is a force conducted through a string, rope, or in this case, a cable, when it is pulled tight by forces acting from opposite ends. It's an example of a force that can act over a distance even though the objects aren't in direct contact. In the context of an elevator, the cable is under tension from the combined forces of the elevator's weight and the deceleration required to bring it to rest.

To calculate this tension, we can refer back to Newton's second law. The tension has to support the weight of the elevator and also account for the additional force needed to decelerate it. Mathematically, this is expressed by \(T = m*a + W\). By substituting the mass (\(m\)), the deceleration (\(a\)), and the weight (\(W\)) into the equation, we can determine the exact tension in the cable at any given moment. It is critical to keep in mind that if the deceleration is directed upwards (as in the elevator problem where it slows the descent), the tension in the cable will be less than the weight of the elevator.