Resonance Practice Problems With Answers

Resonance: Practice Problems with Detailed Answers

Understanding resonance is crucial in physics and engineering, particularly in the study of oscillations and waves. This article provides a comprehensive collection of resonance practice problems, ranging from basic to advanced levels, complete with detailed solutions. We'll cover key concepts like natural frequency, forced oscillations, damping, and the phenomenon of resonance itself, illustrated through various scenarios. By the end, you'll have a firm grasp of resonance and be able to tackle more complex problems with confidence.

I. Introduction to Resonance

Resonance occurs when a system is subjected to an external driving force that matches its natural frequency. At this frequency, the amplitude of the oscillation dramatically increases. Imagine pushing a child on a swing – you get the biggest swing if you push at the right time, matching the swing's natural rhythm. This is resonance in action. The natural frequency depends on the system's physical properties like mass, stiffness, and length. Factors like damping (energy loss due to friction or other resistive forces) influence the sharpness and amplitude of the resonance peak.

II. Practice Problems: Basic Level

Problem 1: A simple pendulum has a length of 1 meter. What is its natural frequency? (Assume g = 9.8 m/s²)

Solution: The natural frequency (f) of a simple pendulum is given by the formula:

f = 1/(2π) * √(g/L)

where:

• g = acceleration due to gravity (9.8 m/s²)
• L = length of the pendulum (1 m)

Substituting the values, we get:

f = 1/(2π) * √(9.8 m/s² / 1 m) ≈ 0.498 Hz

Therefore, the natural frequency of the pendulum is approximately 0.498 Hz.

Problem 2: A mass of 2 kg is attached to a spring with a spring constant of 100 N/m. What is the natural frequency of this simple harmonic oscillator?

Solution: The natural frequency (f) of a mass-spring system is given by:

f = 1/(2π) * √(k/m)

where:

• k = spring constant (100 N/m)
• m = mass (2 kg)

Substituting the values, we get:

f = 1/(2π) * √(100 N/m / 2 kg) ≈ 1.128 Hz

Therefore, the natural frequency of the mass-spring system is approximately 1.128 Hz.

Problem 3: A child's swing acts like a simple pendulum. If a child pushes themselves on the swing every 2 seconds, and the swing has a period of approximately 2 seconds, what phenomenon is at play?

Solution: Since the child's pushing frequency matches the swing's natural period, they are creating resonance. The amplitude of the swing's motion will increase significantly with each push.

III. Practice Problems: Intermediate Level

Problem 4: A damped harmonic oscillator has a natural frequency of 2 Hz and a damping coefficient of 0.5 kg/s. The oscillator is driven by a sinusoidal force with a frequency of 2.1 Hz. Explain what will happen to the amplitude of the oscillation. Will it increase, decrease, or remain constant? Why?

Solution: The driving frequency (2.1 Hz) is close to, but not exactly equal to, the natural frequency (2 Hz). Due to the damping, the amplitude will initially increase but will not reach a very high value. It will eventually settle at a relatively small amplitude because the damping dissipates energy, preventing the system from reaching the full resonance effect it would experience without damping. The amplitude will not remain constant.

Problem 5: Two identical pendulums are set up side-by-side. One pendulum is displaced and released. Over time, the second pendulum starts to oscillate, while the first pendulum’s amplitude decreases. Explain this phenomenon.

Solution: This demonstrates coupled oscillations and energy transfer. Even though the pendulums are not directly connected, slight coupling through air vibrations or a common support allows for the transfer of energy from the first pendulum (initially displaced) to the second pendulum. This energy transfer results in the second pendulum oscillating while the first one loses its amplitude. The system is still subject to damping forces.

Problem 6: A bridge is designed with a natural frequency of 5 Hz. A group of soldiers are marching across the bridge, creating a rhythmic force. At what marching frequency are they most likely to cause dangerous vibrations in the bridge structure, and what concept explains this dangerous effect?

Solution: The soldiers are most likely to cause dangerous vibrations if their marching frequency is close to 5 Hz, the bridge's natural frequency. This is because of resonance. The rhythmic marching forces will significantly amplify the bridge's oscillations, potentially leading to structural damage or collapse if the resonance is strong enough.

IV. Practice Problems: Advanced Level

Problem 7: A driven harmonic oscillator has the equation of motion: m(d²x/dt²) + b(dx/dt) + kx = F₀cos(ωt), where m is the mass, b is the damping coefficient, k is the spring constant, F₀ is the amplitude of the driving force, and ω is the driving frequency. Derive the expression for the amplitude of the steady-state oscillation in terms of these parameters.

Solution: This problem requires solving a second-order differential equation. The steady-state solution (after transient effects have died out) will have the form x(t) = Acos(ωt – φ), where A is the amplitude and φ is the phase difference. The solution involves complex numbers and is beyond the scope of this basic response but it will result in a frequency-dependent amplitude:

A = F₀ / √((k-mω²)² + (bω)²)

Problem 8: Consider a system with two coupled oscillators. Describe how the normal modes of oscillation are related to the concept of resonance.

Solution: A coupled oscillator system has multiple normal modes of oscillation, each with its own characteristic frequency. If an external driving force matches one of these normal mode frequencies, resonance will occur for that specific mode, leading to a large amplitude oscillation in that mode. Other modes will exhibit minimal response.

Problem 9: A wine glass shatters when a high-pitched note is sung near it. Explain the physics behind this.

Solution: The wine glass has a natural frequency determined by its shape and material properties. When a high-pitched note (a sound wave) matches this frequency, resonance occurs. The sound wave's energy is transferred to the glass, causing its vibrations to increase dramatically in amplitude until the glass shatters due to the excessive stress.

V. Explaining Key Concepts

1. Natural Frequency: Every oscillating system (pendulum, mass-spring, etc.) has a natural frequency at which it oscillates freely if disturbed from its equilibrium position. This frequency is determined by the system's physical properties and is independent of external forces.

2. Forced Oscillations: When an external periodic force is applied to an oscillator, it is said to be under forced oscillation. The oscillator will oscillate at the driving frequency of the external force.

3. Damping: Damping is the dissipation of energy from an oscillating system, usually due to friction or resistive forces. It reduces the amplitude of oscillations over time. Without damping, resonance would lead to infinitely increasing amplitudes.

4. Resonance Curve: A resonance curve is a graphical representation of the amplitude of oscillation versus the driving frequency. It shows a peak at the natural frequency, indicating the maximum amplitude of oscillation at resonance. The width of the peak is related to the amount of damping present: higher damping leads to a broader, less sharp peak.

5. Quality Factor (Q-factor): The Q-factor is a dimensionless parameter that measures the sharpness of a resonance. A high Q-factor indicates a narrow, sharp resonance peak, implying low damping and a strong resonance effect. A low Q-factor means a broad, less pronounced peak, representing high damping.

VI. Frequently Asked Questions (FAQ)

Q1: What are some real-world examples of resonance?

• Musical Instruments: The sound produced by musical instruments is due to resonance. The strings, air columns, or membranes vibrate at their natural frequencies.
• Microwave Ovens: Microwaves use resonance to heat food. The microwaves' frequency is chosen to match the resonant frequency of water molecules, causing them to vibrate and generate heat.
• Bridges and Buildings: Resonance is a crucial consideration in structural engineering to avoid catastrophic failures due to externally applied forces that might match the structure's natural frequency.
• MRI Machines: Magnetic Resonance Imaging (MRI) uses resonance in magnetic fields to create detailed images of the human body.

Q2: How does damping affect resonance?

Damping reduces the amplitude of oscillations at resonance. Without damping, the amplitude would increase indefinitely, leading to potential damage or destruction. Damping broadens the resonance curve, making the system less sensitive to frequencies slightly off the natural frequency.

Q3: Can resonance be useful or harmful?

Resonance can be both useful and harmful. Useful applications include musical instruments, microwave ovens, and MRI machines. Harmful effects can occur in structures (bridges, buildings), causing vibrations that exceed their structural limits.

Q4: What is the difference between natural frequency and resonant frequency?

In many simple systems, the natural frequency and resonant frequency are approximately the same. The natural frequency is the frequency at which the system oscillates freely without any external driving force. The resonant frequency is the frequency at which the system responds most strongly to an external driving force. In complex systems with damping or other factors, there can be a slight difference between these two frequencies.

VII. Conclusion

Resonance is a fundamental concept in physics with significant applications in various fields. Understanding its principles – natural frequency, forced oscillations, damping, and the effects of these factors on the amplitude of oscillation – is crucial for tackling a wide range of problems. This article, through a progression of practice problems and explanations, aimed to provide a clear and comprehensive understanding of this critical concept. Remember to always consider the specific context and relevant parameters (mass, stiffness, damping) when analyzing resonance phenomena. By applying these principles, you can accurately predict and analyze the behavior of various oscillating systems.