Question

A lamp hangs vertically from a cord in a descending elevator. The elevator has a deceleration of \(2.4 \mathrm{~m} / \mathrm{s}^{2}\) before coming to a stop. ( \(a\) ) If the tension in the cord is \(89 \mathrm{~N}\), what is the mass of the lamp? (b) What is the tension in the cord when the elevator ascends with an upward acceleration of \(2.4 \mathrm{~m} / \mathrm{s}^{2}\) ?

Short answer

The lamp has a mass of \(7.2 kg\) and when the lift moves upwards, the tension in the cord becomes \(87.84 N\).

Step-by-step solution

Identifying the given quantities and unknown

You have the tension \(T = 89 N\) in the cord and the acceleration due to the elevator's motion \(a = 2.4 m/s^2\). The unknown we're trying to find is the mass \(m\) of the lamp.

Apply Newton's second law to find mass

Newton's second law states that the net force on an object is equal to the mass of the object times the acceleration of the object. In the case of the descending elevator, the net force is the difference between the tension and the weight of the lamp, which gives \(T - mg = -ma\) where \(g\) is the acceleration due to gravity. Solving for \(m\) gives \(m = T / (g + a)\).

Calculations for (a)

Substitute the given values \(T = 89 N, g = 9.8 m/s^2, a = 2.4 m/s^2\) into the equation from step 2 to find \(m = 89 N/ (9.8 m/s^2 + 2.4 m/s^2) = 7.2 kg\).

Finding the tension in upward moving elevator

In the upward moving elevator, the tension acts in the opposite direction to the weight of the object. This gives \(T - mg = ma\), where \(T\) is the tension, \(m\) is the mass of the lamp, \(g\) is acceleration due to gravity, and \(a\) is the acceleration of the elevator. Do the math to calculate the new tension.

Calculations for (b)

Substitute the calculated mass \(m = 7.2 kg\), acceleration due to gravity \(g = 9.8 m/s^2\), acceleration of the elevator \(a = 2.4 m/s^2\) in the equation to get \(T = m * (g + a) = 7.2 kg * (9.8 m/s^2 + 2.4 m/s^2) = 87.84 N\).

Key concepts

Understanding Tension in a Cord

The tension in a cord is a force that acts through a string, rope, or any other type of cable. In our exercise, the tension (\( T \)) in the cord holding the lamp relates to the forces acting when the elevator moves. Tension must balance the force of gravity acting on the lamp and the additional force due to the elevator's acceleration.
To think of it another way:

• When the elevator descends and decelerates, tension reduces to balance against gravity and deceleration.
• When the elevator ascends with upward acceleration, tension has to counter both gravity and this added motion, increasing its value.

The idea is to understand how tension changes based on the elevator's motion. If the elevator were static, tension would simply equal the lamp's weight. In motion, we adjust it according to the balance of forces using Newton's Second Law.

Grasping Acceleration

Acceleration is all about change. It's how quickly speed changes over time. In elevator physics, we look at both ascending and descending motions.
The exercise gives us a specific value for acceleration, \( 2.4 \ \text{m/s}^2 \), whether the elevator descends or ascends. By considering the effect of this acceleration on the forces acting on the lamp, we can determine how the tension in the cord will need to adjust.
Remember:

• A descending elevator slows the lamp's fall, meaning less tension is required than if the elevator were still.
• Ascending pushes the lamp against gravity, increasing tension.

So, when studying acceleration in terms of tension, consider the elevator's direction and speed change nature.

Exploring Mass Calculation

To calculate mass using tension and acceleration, we rely on Newton's Second Law, \( F = ma \). Solving these problems involves separating forces acting on the lamp:

• The force of gravity \( mg \).
• The net force due to elevator's motion (tension involved).

To find mass (\( m \)), we rearrange the known values in our equation. So, using the formula \( m = \frac{T}{g+a} \) for a descending elevator, we determine the lamp's mass by dividing tension over combined gravitational and elevator's acceleration.
It's a simple yet powerful calculation which can be applied to solve various physics exercises that involve moving systems like elevators.

Demystifying Elevator Physics

Elevator physics deals with how objects inside these moving systems experience forces differently compared to being stationary. The motion of an elevator alters how forces act on objects inside.
In our exercise:

• When descending or ascending, the elevator's motion changes how we perceive forces such as tension.
• It alters the balance between forces of gravity and tension, due to either slowing down or speeding up.

Understanding these shifts is crucial. When the elevator moves downwards, the tensions lessen because the downward force aids gravity. Conversely, when soaring upwards, tension must rise to combat gravity.
This is why elevator physics is a fascinating topic, often appearing in physics problems concerning forces and motion.