define-period-frequency-displacement-as-a-function-of-time
🚀 In physics, oscillations and waves are fundamental concepts that describe repetitive motions and the transfer of energy through space. Periodic motion refers to any motion that repeats itself at regular intervals, while harmonic motion is a specific type of periodic motion that follows a sine or cosine function. The period of a motion is the time taken to complete one full cycle, and frequency is the number of cycles completed in one second. Displacement as a function of time describes how the position of an oscillating object changes over time, typically represented by a sinusoidal function.
Theory Explanation
Understanding Period and Frequency
The period (T) is the time taken for one complete cycle of motion. It is measured in seconds. Frequency (f) is the number of cycles per second and is measured in Hertz (Hz). The relationship between period and frequency is given by the formula: f = 1/T.
Displacement as a Function of Time
For harmonic motion, the displacement (x) of an object can be described as a function of time (t) using the equation: x(t) = A \cos(\omega t + \phi), where A is the amplitude, \omega is the angular frequency, and \phi is the phase constant. This equation shows how the position of the object changes over time in a periodic manner.
Key Points
- 🎯 The period is the time for one complete cycle of motion.
- 🎯 Frequency is the number of cycles per second, inversely related to the period.
- 🎯 Displacement in harmonic motion can be modeled using cosine or sine functions.
- 🎯 Amplitude represents the maximum displacement from the equilibrium position.
- 🎯 Angular frequency relates to the frequency and is given by \( \omega = 2\pi f \).
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Examples:💡
A mass on a spring oscillates with a period of 2 seconds. Calculate its frequency and write the displacement function if the amplitude is 5 cm and the phase constant is 0.
Solution:
Step 1: Calculate the frequency using the period: f = 1/T = 1/2 = 0.5 Hz.
Step 2: Write the displacement function using the given amplitude and phase constant: x(t) = 5 \cos(\omega t + 0).
Common Mistakes
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Mistake: Confusing period and frequency; students often mix up the definitions.
Correction: Remember that period is the time for one cycle, while frequency is how many cycles occur in one second.
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Mistake: Incorrectly applying the displacement function; students may forget to include the phase constant or use the wrong trigonometric function.
Correction: Always check the parameters of the motion and ensure the correct function (sine or cosine) is used based on the initial conditions.