Calculating Tension in a Hanging Object in an Elevator Slowing Upward

Introduction

Understanding the forces at play when a hanging object is moved within an accelerating or decelerating environment is a fundamental concept in physics. This article will explore the tension in a string supporting an object inside an elevator that is moving upwards but slowing down. We will apply Newton's second law to derive the tension in the string.

Understanding the Forces Acting on the Object

Let's consider an object hanging by a string from the ceiling of an elevator. The object will experience two main forces: gravity (downward) and the tension in the string (upward).

Weight (W)

The downward force due to gravity is given by:

W mg, where m is the mass of the object and g is the acceleration due to gravity.

Note: g ≈ 9.81 m/s2

Tension (T)

The upward force exerted by the string on the object is known as the tension.

Thermal Physics of Elevator Motion

When the elevator is moving upward but slowing down, it experiences a downward acceleration a. This deceleration means that the net acceleration of the object is downward, even though the elevator itself is moving upward.

Applying Newton's Second Law

Newton's second law states that the net force acting on an object is the mass of the object times its acceleration:

F_{net} ma

The net force on the object is the difference between the tension in the string and the object's weight:

F_{net} T - W

Substituting the expressions for F_{net} and W:

T - mg -ma

Rearranging this equation to solve for the tension T:

T mg - ma

Factoring out m gives:

T m(g - a)

Conclusion and Calculation

The tension in the string when the elevator is slowing up is given by:

T m(g - a)

g ≈ 9.81 m/s2 (acceleration due to gravity) a is the magnitude of the elevator's downward deceleration.

This formula shows that the tension in the string is less than the weight of the object when the elevator is decelerating while moving upward.

Example Calculation

Let's consider an example where the mass of the object is 5.0 kg and the elevator has a downward acceleration of 2.0 m/s2.

Using the formula:

T 5.0 * (9.81 - 2.0) or T 5.0 * 7.81 39.05 N

Rounding to two significant figures, the tension is approximately T ≈ 39 N.

Conclusion

Understanding the forces and the application of Newton's second law in the context of a decelerating elevator provides insight into the behavior of objects in non-inertial frames. The tension in the string can be calculated precisely using the provided formula.