When a mass is attached to a vertical spring, the spring is...

When a mass is attached to a vertical spring, the spring is stretched a distance $d$. The mass is then pulled down from this position and released. It undergoes 50.0 ascillations in $30.0 \mathrm{~s}$. What was the distance $d \overrightarrow{2}$

Key Concepts

Simple Harmonic Motion

This concept describes the type of oscillatory motion in which the restoring force is directly proportional to the displacement from an equilibrium position, resulting in a motion that repeats sinusoidally over time. In the mass?spring system, once the mass is displaced from equilibrium and released, it undergoes simple harmonic motion.

Oscillation Period and Frequency

The oscillation period is the time taken for one complete cycle of motion, and the frequency is the number of cycles per unit time. These quantities are interrelated, as knowing the number of oscillations over a given time allows determination of the period. For a mass-spring system undergoing simple harmonic motion, the period is given by T = 2??(m/k), which links the mass and the spring constant.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its displacement from the unstressed position, mathematically expressed as F = -kx, where k is the spring constant and x is the displacement. This law is fundamental in analyzing the forces in a vertical spring system, especially at equilibrium when the spring's restoring force balances the gravitational force.

Equilibrium Extension

When a mass is attached to a vertical spring and allowed to come to rest, it stretches until the spring’s restoring force equals the gravitational force acting on the mass. This equilibrium condition, expressed as mg = kd, allows the determination of the static extension d of the spring due to the weight of the mass, which is crucial for understanding the subsequent oscillatory motion when the mass is further displaced.

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