A spring with spring constant 15 N / m hangs from the...

A spring with spring constant $15 \mathrm{N} / \mathrm{m}$ hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down $6.0 \mathrm{cm}$ and released. If the ball makes 30 oscillations in $20 \mathrm{s}$, what are its (a) mass and (b) maximum speed?

Key Concepts

Maximum Speed in SHM

In simple harmonic motion, the maximum speed occurs as the system passes through the equilibrium position and is determined by the product of the amplitude and angular frequency (v_max = A?). This relationship shows how the initial displacement and the inherent properties of the system combine to influence the energy and dynamics of the motion.

Angular Frequency and Oscillation Period

Angular frequency is defined as ? = ?(k/m) for a mass-spring system and represents the rate of oscillation in radians per second. The oscillation period, given by T = 2?/?, is the time for one complete cycle, linking the system’s physical properties to its temporal behavior.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, with the spring constant being the proportionality factor. This law is fundamental for analyzing how springs behave under load and sets the stage for understanding mass-spring systems.

Simple Harmonic Motion (SHM)

SHM describes a type of oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium. This results in sinusoidal oscillations, and the motion is predictable and periodic, making SHM a foundational concept in understanding vibrational systems.

Mass-Spring System Dynamics

A mass-spring system, where a mass is attached to a spring, is a classic example of a system undergoing simple harmonic motion. The properties of the system such as the mass and spring constant determine the oscillation period and frequency, providing insights into how system parameters affect motion.

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