What are the (a) amplitude, (b) frequency, and (c) phase constant of the oscillation shown in FIGURE EX15.6?

Verified step by step guidance

1. Identify the amplitude of the oscillation by measuring the maximum displacement from the equilibrium position. In this graph, the maximum displacement is 5 cm.
2. Determine the period of the oscillation by measuring the time it takes for one complete cycle. From the graph, one complete cycle occurs from t = 0 s to t = 4 s.
3. Calculate the frequency of the oscillation using the formula f = 1/T, where T is the period. Here, T = 4 s.
4. Identify the phase constant by examining the initial condition of the oscillation. At t = 0 s, the displacement is 2 cm. Use the general form of the equation for simple harmonic motion, x(t) = A cos(ωt + φ), to solve for the phase constant φ.
5. Use the angular frequency ω, which is related to the frequency by ω = 2πf, to complete the equation for the oscillation.

Key Concepts

Amplitude

Amplitude is the maximum displacement of an oscillating object from its equilibrium position. In the context of simple harmonic motion, it represents the peak value of the oscillation, indicating how far the object moves from the center point. In the provided graph, the amplitude can be determined by measuring the maximum vertical distance from the equilibrium line (x=0) to the highest or lowest point of the wave.

Frequency

Frequency is the number of complete cycles of oscillation that occur in a unit of time, typically measured in hertz (Hz). It is inversely related to the period of the oscillation, which is the time taken for one complete cycle. To find the frequency from the graph, one can count the number of cycles within a given time interval and divide by that time, providing insight into how quickly the oscillation occurs.

Phase Constant

The phase constant is a parameter that indicates the initial angle or position of the oscillating object at time t=0. It helps define the starting point of the oscillation in relation to the sine or cosine function used to describe simple harmonic motion. In the graph, the phase constant can be inferred from the position of the wave at the beginning of the time interval, determining how the wave is shifted horizontally from a standard sine or cosine wave.