How to Calculate Resonance Frequency: Step-by-Step Examples

📚 Quick Study Guide

• 🍎 Resonance frequency is the natural frequency at which an object vibrates with maximum amplitude.
• ➗ For a simple harmonic oscillator (like a mass-spring system), the resonance frequency ($f$) is given by: $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$, where $k$ is the spring constant and $m$ is the mass.
• ⚡ In an LCR circuit (inductor, capacitor, resistor), the resonance frequency is: $f = \frac{1}{2\pi \sqrt{LC}}$, where $L$ is the inductance and $C$ is the capacitance.
• 💡 At resonance, the impedance in an LCR circuit is at its minimum, and the current is at its maximum.
• 📐 Units are crucial! Ensure consistent units (e.g., kg for mass, N/m for spring constant, Henry for Inductance, Farad for Capacitance).

Practice Quiz

1. Question 1: What is the formula for resonance frequency in a simple harmonic oscillator?
1. $f = 2\pi \sqrt{\frac{k}{m}}$
2. $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$
3. $f = \frac{1}{2\pi} \sqrt{\frac{m}{k}}$
4. $f = 2\pi \sqrt{\frac{m}{k}}$
2. Question 2: In an LCR circuit, what parameters determine the resonance frequency?
1. Resistance and Capacitance
2. Inductance and Resistance
3. Inductance and Capacitance
4. Resistance, Inductance, and Capacitance
3. Question 3: What happens to the impedance in an LCR circuit at resonance?
1. It is at its maximum.
2. It is at its minimum.
3. It remains constant.
4. It becomes infinite.
4. Question 4: If the spring constant of a spring-mass system is 100 N/m and the mass is 1 kg, what is the resonance frequency?
1. 1.59 Hz
2. 4.00 Hz
3. 10.0 Hz
4. 0.159 Hz
5. Question 5: An LCR circuit has an inductance of 1 H and a capacitance of 1 μF. What is the resonance frequency?
1. 159 Hz
2. 1.59 kHz
3. 1.59 Hz
4. 15.9 kHz
6. Question 6: What is the SI unit for inductance?
1. Farad
2. Ohm
3. Henry
4. Tesla
7. Question 7: Which of the following changes would increase the resonance frequency of a mass-spring system?
1. Increasing the mass.
2. Decreasing the spring constant.
3. Increasing the spring constant.
4. Increasing both mass and spring constant proportionally.