Simple Harmonic Motion (SHM): The Science of Oscillations
What Is Simple Harmonic Motion?
SHM describes periodic oscillations where the restoring force is proportional to displacement and directed toward equilibrium.
Key Equations in SHM
Displacement (xxx)
The position of the object at time ttt:
x=Acos(ωt+ϕ)x = A \cos(\omega t + \phi)x=Acos(ωt+ϕ)
Where:
- AAA: Amplitude.
- ω\omegaω: Angular frequency (rad/srad/srad/s).
- ϕ\phiϕ: Phase constant.
Velocity (vvv)
The rate of change of displacement:
v=−ωAsin(ωt+ϕ)v = -\omega A \sin(\omega t + \phi)v=−ωAsin(ωt+ϕ)
Acceleration (aaa)
Proportional to displacement:
a=−ω2xa = -\omega^2 xa=−ω2x
Energy in SHM
Kinetic Energy (KEKEKE)
Energy due to motion:
KE=12mω2(A2−x2)KE = \frac{1}{2} m \omega^2 (A^2 – x^2)KE=21mω2(A2−x2)
Potential Energy (PEPEPE)
Energy due to position:
PE=12mω2x2PE = \frac{1}{2} m \omega^2 x^2PE=21mω2×2
Applications of SHM
Clocks
Pendulums use SHM to maintain consistent timekeeping.
Engineering
SHM principles are applied in suspension systems and shock absorbers.
Medical Imaging
Ultrasound machines rely on oscillating sound waves.
Example Problem
A 2 kg2 \, \text{kg}2kg mass oscillates on a spring with a spring constant of 50 N/m50 \, \text{N/m}50N/m and amplitude of 0.1 m0.1 \, \text{m}0.1m. Find the period and maximum velocity.
- Angular Frequency (ω\omegaω):
ω=km=502=5 rad/s\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{2}} = 5 \, \text{rad/s}ω=mk=250=5rad/s
- Period (TTT):
T=2πω=2π5≈1.26 sT = \frac{2\pi}{\omega} = \frac{2\pi}{5} \approx 1.26 \, \text{s}T=ω2π=52π≈1.26s
- Maximum Velocity (vmaxv_{\text{max}}vmax):
vmax=ωA=5⋅0.1=0.5 m/sv_{\text{max}} = \omega A = 5 \cdot 0.1 = 0.5 \, \text{m/s}vmax=ωA=5⋅0.1=0.5m/s
Common Mistakes in SHM Problems
- Forgetting the negative sign in the restoring force equation.
- Mixing up displacement and amplitude.
- Confusing angular frequency with regular frequency.
Practice Questions
- A 1 kg1 \, \text{kg}1kg mass oscillates with an amplitude of 0.2 m0.2 \, \text{m}0.2m and a spring constant of 25 N/m25 \, \text{N/m}25N/m. Find the maximum velocity.
- Explain how energy is conserved in SHM.
- Describe one real-world example of SHM in engineering.