Understanding Elevator Acceleration with a Spring Balance: A Physics Puzzle

Imagine a scenario where a passenger is inside an elevator that suddenly starts accelerating upward. The passenger notices that a 1 kg mass, placed on a spring balance inside the elevator, now registers 1.2 kg. This observation raises an important question: what is the acceleration of the elevator?

The Role of a Spring Balance in Measuring Force

Firstly, it is important to clarify that a spring balance measures force, not mass. The reading of 1.2 kg on the balance actually refers to the force exerted on the spring, which is equivalent to the weight of 1.2 kg under standard Earth gravity. The misunderstanding often arises from the common practice of equating force (in newtons) to the equivalent mass under Earth's gravity, but the balance itself is measuring the force directly, not the mass.

Applying Mechanics to Solve the Problem

The key to solving this problem lies in understanding the principle of dynamical equilibrium. When the elevator accelerates upward, it exerts an extra force on the mass that is placed inside, in addition to the gravitational force. The net force acting on the mass can be calculated using Newton's second law of motion, which states that the net force (F) is equal to the mass (m) times the acceleration (a): F ma.

Calculating the Net Force

Let's break down the forces acting on the mass:

Force due to Earth's gravity: 1 kg * 9.8 m/s2 9.8 N (downward) Force exerted by the spring balance: 1.2 kg * 9.8 m/s2 11.76 N (upward, as the balance reads 12 N)

Now, we need to find the net force acting on the mass:

Net force 11.76 N (upward) - 9.8 N (downward) 1.96 N (upward)

Computing the Acceleration

With the net force known, we can now use Newton's second law to find the acceleration:

F ma

1.96 N 1 kg * a

Therefore, a 1.96 m/s2 (upward)

This means that the elevator is accelerating upward at a rate of approximately 1.96 m/s2, which is approximately 20% of Earth's gravitational acceleration (9.8 m/s2).

Conclusion

The spring balance reading of 1.2 kg translates to a force of 11.76 N, which combined with Earth's gravity results in a net force of 1.96 N acting on the mass. Using Newton's second law, we can calculate the acceleration of the elevator to be approximately 1.96 m/s2, or about 0.2 times Earth's gravitational acceleration. This explanation helps to illuminate the interaction between forces, mass, and acceleration in a moving elevator.