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Oscillations: Exploring Simple Harmonic Motion and Damped Systems
What Are Oscillations?
Oscillations are periodic motions about an equilibrium position, such as a pendulum or a mass on a spring.
Key Equations in Simple Harmonic Motion (SHM)
Displacement (\( x \))
Position as a function of time:
\[ x = A\cos(\omega t + \phi) \]
Where:
- \( A \): Amplitude (m)
- \( \omega \): Angular frequency (rad/s)
- \( \phi \): Phase constant
Velocity (\( v \)) and Acceleration (\( a \))
\[ v = -\omega A\sin(\omega t + \phi) \]
\[ a = -\omega^2 x \]
Energy in SHM
Total energy is constant and shared between kinetic and potential forms:
\[ E = \frac{1}{2}kA^2 \]
Damped Oscillations
Types of Damping
- Light Damping: Gradual energy loss (e.g., pendulum in air)
- Critical Damping: Fast return to equilibrium without oscillating (e.g., car suspensions)
- Overdamping: Slower return to equilibrium (e.g., heavy damping)
Applications of Oscillations
Engineering
SHM principles are used in suspension systems and shock absorbers.
Clocks
Pendulums maintain precise timekeeping.
Medicine
Ultrasound machines rely on oscillatory motion of sound waves.
Example Problem
A mass (\( 2 \, \text{kg} \)) on a spring (\( k = 50 \, \text{N/m} \)) oscillates with an amplitude of \( 0.1 \, \text{m} \). Find the period and maximum velocity.
- Angular Frequency (\( \omega \)):
\[ \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{2}} = 5 \, \text{rad/s} \] - Period (\( T \)):
\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{5} \approx 1.26 \, \text{s} \] - Maximum Velocity (\( v_{\text{max}} \)):
\[ v_{\text{max}} = \omega A = 5 \times 0.1 = 0.5 \, \text{m/s} \]
Common Mistakes in SHM Calculations
- Forgetting to square the angular frequency in acceleration
- Ignoring the direction of velocity and acceleration vectors
- Mixing up energy forms during oscillations
Practice Questions
- A spring (\( k = 25 \, \text{N/m} \)) oscillates with a mass of \( 1 \, \text{kg} \). Calculate the period of oscillation.
- Explain the differences between light and critical damping.
- Describe one application of SHM in medical imaging.