A block attached to a spring with unknown spring constant oscillates with a period of 2.0 s. What is the period if c. The amplitude is doubled?

Verified step by step guidance

1. Identify that the period of oscillation for a mass-spring system is given by the formula $T = 2\pi\sqrt{\frac{m}{k}}$, where $T$ is the period, $m$ is the mass, and $k$ is the spring constant.
2. Recognize that the amplitude of the oscillation does not appear in the formula for the period. This indicates that the period of a simple harmonic oscillator depends only on the mass and the spring constant, not on the amplitude.
3. Understand that doubling the amplitude of the oscillation does not affect the mass of the block or the spring constant of the spring.
4. Conclude that since the period $T$ depends only on $m$ and $k$, and neither of these parameters is changed when the amplitude is doubled, the period remains the same.
5. State that the period of the oscillation will remain 2.0 s even if the amplitude is doubled.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion is characterized by a restoring force proportional to the displacement from that position, leading to a sinusoidal time dependence. In the case of a mass-spring system, the period of oscillation is determined by the mass and the spring constant, but not by the amplitude.

Period of Oscillation

The period of oscillation is the time taken for one complete cycle of motion in a periodic system. For a mass-spring system undergoing SHM, the period (T) is given by the formula T = 2π√(m/k), where m is the mass and k is the spring constant. Importantly, the period remains constant regardless of the amplitude of oscillation, as long as the system is not subject to non-linear effects.

Amplitude

Amplitude refers to the maximum extent of displacement from the equilibrium position in oscillatory motion. In SHM, while the amplitude affects the energy of the system, it does not influence the period of oscillation. Therefore, even if the amplitude is doubled, the period of the oscillation remains unchanged, reaffirming the independence of period from amplitude in ideal conditions.