4.22 Apparent Weight in an Elevator

Introduction

An object's real weight is the gravitational force acting on it, given by $W=mg$. However, the weight we feel or measure with a scale is actually the contact force, such as a normal force or tension, that supports us. This measured force is called apparent weight. In an accelerating frame of reference, such as an elevator, the apparent weight can differ from the real weight. This phenomenon provides a clear application of Newton's Second Law of Motion.

Analyzing the Forces

Consider a person of mass $m$ standing on a weighing scale inside an elevator. The scale measures the normal force $N$. The same analysis applies if the person is hanging from a spring balance, which measures tension $T$. We examine four scenarios to determine the apparent weight in each case.

Case 1: Elevator at Rest or Moving with Constant Velocity

Condition: The elevator's acceleration is zero ($a=0$).

Analysis: Since there is no acceleration, the net force on the person is zero. The upward normal force $N$ must balance the downward force of gravity $W=mg$.

Applying Newton's Second Law: $N-mg=ma=m(0)=0$

Result: $N=mg$

In this case, the apparent weight equals the real weight.

Case 2: Elevator Accelerating Upwards

Condition: The elevator accelerates upwards with acceleration $a$.

Analysis: To accelerate the person upwards, there must be a net upward force. The upward normal force from the scale must be greater than the downward force of gravity.

Applying Newton's Second Law: $N-mg=ma$

Result: $N=mg+ma=m(g+a)$

The apparent weight is greater than the real weight. The person feels "heavier."

Case 3: Elevator Accelerating Downwards

Condition: The elevator accelerates downwards with acceleration $a$.

Analysis: To accelerate the person downwards, the net force must be directed downward. The downward force of gravity exceeds the upward normal force from the scale.

Applying Newton's Second Law: $mg-N=ma$

Result: $N=mg-ma=m(g-a)$

The apparent weight is less than the real weight. The person feels "lighter."

Case 4: Elevator in Free Fall

Condition: The elevator cable snaps, and it falls freely. The downward acceleration equals the acceleration due to gravity ($a=g$).

Analysis: This is a special case of downward acceleration. The person and the scale fall at the same rate.

Applying the formula from Case 3: $N=m(g-a)$

Substitute $a=g$: $N=m(g-g)$

Result: $N=0$

The apparent weight is zero. The scale reads zero, and the person experiences the sensation of weightlessness.

Summary of Cases

Possible Questions and Answers

Q: What is the difference between real weight and apparent weight?

A: Real weight is the force of gravity on an object ($mg$). Apparent weight is the force an object exerts on its support, which is what a scale measures. Apparent weight only equals real weight when there is no vertical acceleration.

Q: What would happen if an elevator accelerated downwards faster than $g$?

A: If $a>g$, the formula $N=m(g-a)$ yields a negative result. This is physically impossible for a scale. The person would lift off the floor and be pinned to the ceiling of the elevator as it accelerates downward faster than natural free fall.