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 (a) from x = 0.180 m to x = -0.180 m?

Verified step by step guidance

1. Identify the type of motion: The block is moving in Simple Harmonic Motion (SHM), which is characterized by its sinusoidal motion and dependence on the spring's properties.
2. Understand the relationship between amplitude and period in SHM: The period of SHM is independent of the amplitude. This means that even if the amplitude changes, the period of the oscillation remains the same.
3. Recall the definition of period in SHM: The period (T) is the time it takes for the block to complete one full cycle of motion, from one extreme to the other and back again.
4. Apply the information given in the problem: Since the period is independent of amplitude, the time taken to travel from x = 0.180 m to x = -0.180 m will be the same as the time taken to travel from x = 0.090 m to x = -0.090 m, which is given as 2.70 seconds.
5. Conclude that the time taken for the block to travel from x = 0.180 m to x = -0.180 m when the amplitude is doubled is also 2.70 seconds.

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 the equilibrium, leading to sinusoidal motion. In SHM, the period and frequency are constant, depending only on the system's properties, such as mass and spring constant.

Amplitude

Amplitude refers to the maximum extent of displacement from the equilibrium position in oscillatory motion. In the context of SHM, it is the maximum distance the block moves from its rest position. Doubling the amplitude increases the distance the block travels during each oscillation, but it does not affect the period of the motion, which remains constant for a given system.

Period of Oscillation

The period of oscillation is the time taken for one complete cycle of motion in SHM. It is independent of the amplitude for ideal springs and is determined by the mass of the block and the spring constant. The formula for the period (T) in SHM is T = 2π√(m/k), where m is the mass and k is the spring constant, indicating that changes in amplitude do not affect the period.