Section 6.2: Period, Frequency, and Amplitude
In SHM, the period (T) is the time to complete one full oscillation, the frequency (f) is the number of oscillations per second, and the amplitude (A) is the maximum displacement from equilibrium.
Formulas:
- \( T = \frac{2\pi}{\omega} \)
- \( f = \frac{1}{T} \)
- \( x_\text{max} = A \) (Amplitude)
Relationship between displacement, velocity, and acceleration:
- \( x(t) = A \cos(\omega t + \phi) \)
- \( v(t) = -\omega A \sin(\omega t + \phi) \)
- \( a(t) = -\omega^2 A \cos(\omega t + \phi) \)
Example 1
A mass on a spring oscillates with amplitude 0.2 m and angular frequency 5 rad/s. Find the period, frequency, and maximum speed.
\( T = 2\pi/\omega = 2\pi/5 \approx 1.26 \text{ s} \)
\( f = 1/T \approx 0.793 \text{ Hz} \)
\( v_\text{max} = \omega A = 5 \cdot 0.2 = 1 \text{ m/s} \)
Practice Problems
- A mass-spring system oscillates with frequency 2 Hz. Find the period.
- An oscillator has amplitude 0.1 m and angular frequency 10 rad/s. Find maximum velocity.
- A pendulum has period 1.5 s. Determine its frequency.
- Displacement in SHM is \(x = 0.05 \cos(20t)\). Find period and frequency.
- A mass oscillates with amplitude 0.3 m and frequency 0.5 Hz. Find maximum acceleration.