Question

An elevator weighing $6200\mathrm{lb}$ is pulled upward by a cable with an acceleration of $3.8\mathrm{ft}/{\mathrm{s}}^2$. (a) What is the tension in the cable? (b) What is the tension when the elevator is accelerating downward at $3.8\mathrm{ft}/{\mathrm{s}}^2$ but is still moving upward?

Short answer

The tension in the cable when the elevator is pulled upward is 6300 lb, while the tension when the elevator is accelerating downward but still moving upward is 6100 lb.

Step-by-step solution

Define the Forces

In this situation, there are two forces to consider: the force due to gravity, which pulls the elevator downward, and the tension on the cable which either pulls it upwards or opposes downward motion. The weight of the elevator can be found by multiplying its mass by the acceleration due to gravity. However, the weight is given in pounds, which is a unit of force, so this step is already done for us - the weight is 6200 lb.

Calculate Tension when Elevator is Pulled Upward

When the elevator is pulled upward with an acceleration of 3.8 ft/s², the tension in the cable must overcome both the weight of the elevator and provide the upward force necessary to accelerate it. This can be calculated by using Newton's second law in the form $F=ma$, where $F$ is the sum of the forces on the elevator, $m$ is its mass, and $a$ is its acceleration. The net force can also be written as the tension minus the weight, so we have: $T-W=ma$. Solving for $T$, we get: $T=W+ma=6200lb+(6200lb/32.2ft/s²)\ast 3.8ft/s²$, where we've divided the weight by the acceleration due to gravity (32.2 ft/s²) to get the mass. This gives $T=6300lb$.

Calculate Tension when Elevator is Accelerating Downward

When the elevator is accelerating downward at 3.8 ft/s² but is still moving upward, the tension in the cable has to overcome the downward acceleration but does not need to support the full weight of the elevator. This situation is represented by the equation: $T=W-ma=6200lb-(6200lb/32.2ft/s²)\ast 3.8ft/s²$. This gives $T=6100lb$.

Key concepts

Newton's second law

In the context of an elevator tension problem, understanding Newton's second law is critical. This law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration, which is succinctly expressed as $F=ma$. In practical terms, when an object is accelerating, there must be a net force acting on it.

For our elevator, this law elucidates how the cable's tension must sufficiently counter the force due to gravity while also providing the extra oomph - that force needed for the upward acceleration of the elevator. If you imagine momentarily that the elevator is in freefall, the only force at play is gravity pulling it down. Now add the cable's tension pulling up, and you get the elevator accelerating upwards as the net force tilts in that direction. In the reverse case with the elevator accelerating downwards, that tension lessens because it now supports less than the elevator's full weight since the direction of the cable's tension force and gravity's pull are the same.

force calculation

Force calculations are the bread and butter of physics problems, especially when dealing with elevator scenarios. The first step is always to identify all the forces at play - in this case, gravity (weight) and cable tension. The weight of the elevator is given directly, which simplifies our task. We then establish the relationship between the forces using Newton's second law.

To be precise, for an upward acceleration, the tension must be stronger than the weight to result in a net force that can accelerate the elevator upwards. Mathematically, this translates to the tension $T$ equaling the weight plus the product of mass and acceleration. Conversely, when the elevator accelerates downwards, the tension is reduced by the same product of mass and acceleration, since the acceleration due to gravity is in the same direction as the cable's tension.

acceleration and tension

When learning about acceleration and tension, especially within elevator problems, one must acknowledge the direct relationship between them. The greater the acceleration required, the greater the tension in the cable must be, provided the direction of acceleration is upwards. This is because the tension needs to not only balance the force of gravity but also contribute to the upward motion.

However, this relationship flips when the elevator accelerates in the same direction as gravity. As acceleration increases downward, the tension decreases since it needs to overcome a smaller portion of the gravitational force. The essence of solving these problems lies in understanding how to reconcile the forces involved with the intended motion of the elevator. In both scenarios, be it upward or downward acceleration, the tension is directly influenced by the magnitude and direction of acceleration, showing how intertwined these two concepts are.