An object is undergoing SHM with period 0.900 s and amplitude 0.320 m. At t = 0 the object is at x = 0.320 m and is instantaneously at rest. Calculate the time it takes the object to go (a) from x = 0.320 m to x = 0.160 m. (b) from x = 0.160 m to x = 0.

Verified step by step guidance

1. Identify the angular frequency ($\omega$) using the formula $\omega =\frac{2\pi }{T}$, where $T$ is the period of the simple harmonic motion (SHM).
2. Recognize that the object starts at its maximum amplitude ($x=A$) and is momentarily at rest, indicating it is at one of the extreme points of its motion.
3. Use the SHM position function $x(t)=A\cos (\omega t)$ to describe the motion of the object, where $A$ is the amplitude.
4. To find the time it takes to go from $x=0.320m$ to $x=0.160m$, solve the equation $0.160=0.320\cos (\omega t)$ for $t$.
5. To find the time it takes to go from $x=0.160m$ to $x=0m$, solve the equation $0=0.320\cos (\omega t)$ for $t$, and subtract the time found in the previous step from this result.

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. Key parameters include amplitude, period, and frequency, which define the motion's characteristics.

Period and Amplitude

The period of an oscillating object is the time taken to complete one full cycle of motion, measured in seconds. Amplitude refers to the maximum displacement from the equilibrium position. In this case, the period is 0.900 s, and the amplitude is 0.320 m, indicating how far the object moves from its rest position during oscillation.

Phase of Motion

The phase of motion in SHM describes the position of the object at a specific time relative to its cycle. At t = 0, the object is at its maximum displacement (0.320 m) and at rest, indicating it is at the peak of its motion. Understanding the phase helps in calculating the time taken to move between different positions during the oscillation.