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

Verified step by step guidance

1. Step 1: Recall the formula for the period of a spring-mass system: $T=2\pi \surd \frac{m}{k}$, where $T$ is the period, $m$ is the mass, and $k$ is the spring constant.
2. Step 2: Understand that the period of oscillation for a spring-mass system depends only on the mass and the spring constant. It does not depend on the amplitude of oscillation.
3. Step 3: Since the amplitude is doubled, this change does not affect the period of the oscillation. The period remains the same as long as the system is undergoing simple harmonic motion.
4. Step 4: Confirm that the given period of 2.0 s will remain unchanged regardless of the amplitude change.
5. Step 5: Conclude that the period of the oscillation is still 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 ideal and not subject to damping or 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 (more amplitude means more potential and kinetic energy), it does not influence the period of oscillation. Therefore, doubling the amplitude of the oscillation does not change the period, which remains at 2.0 seconds in this scenario.