A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is 0.090 m, it takes the block 2.70 s to travel from x = 0.090 m to x = -0.090 m. If the amplitude is doubled, to 0.180 m, how long does it take the block to travel from x = 0.090 m to x = -0.090 m?

Verified step by step guidance

1. Understand that the block is undergoing Simple Harmonic Motion (SHM), which is characterized by oscillations around an equilibrium position. The time taken for one complete cycle is the period of the motion.
2. Recall that the period of SHM is independent of the amplitude. The period \( T \) is given by \( T = \frac{2\pi}{\omega} \), where \( \omega \) is the angular frequency.
3. Given that the block takes 2.70 s to travel from \( x = 0.090 \) m to \( x = -0.090 \) m, this time represents half of the period, as it covers half of the oscillation cycle.
4. Since the period is independent of amplitude, doubling the amplitude to 0.180 m does not change the period. Therefore, the time to travel from \( x = 0.090 \) m to \( x = -0.090 \) m remains the same.
5. Conclude that the time taken for the block to travel from \( x = 0.090 \) m to \( x = -0.090 \) m is still 2.70 s, even if the amplitude is doubled.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. In SHM, the motion is sinusoidal, characterized by a constant frequency and amplitude, and is often exemplified by systems like a mass-spring system or a pendulum.

Amplitude in SHM

Amplitude in SHM refers to the maximum displacement of the object from its equilibrium position. It is a measure of the energy in the system, with larger amplitudes indicating more energy. In the context of the problem, changing the amplitude affects the total energy but does not alter the period of the motion, as the period is independent of amplitude in ideal SHM.

Period and Frequency in SHM

The period in SHM is the time taken for one complete cycle of motion, while frequency is the number of cycles per unit time. For a mass-spring system, the period is determined by the mass and the spring constant, and is independent of amplitude. Thus, even if the amplitude changes, the time taken for specific displacements remains constant, as the period does not change.