Oscillations and Waves: NMAT Physics Review

Oscillations and waves are core topics in NMAT Physics, testing both conceptual understanding and mathematical problem-solving skills. These topics frequently appear in questions related to motion, energy, sound, light, and modern applications in medicine and engineering. A strong grasp of oscillatory motion and wave behavior will help you solve a wide range of NMAT problems efficiently.

Introduction to Oscillatory Motion

Oscillatory motion refers to a type of motion in which an object repeatedly moves back and forth about a fixed equilibrium position. This motion is characterized by periodic repetition and is commonly observed in systems such as pendulums, springs, vibrating strings, and atoms in a solid lattice.

Key features of oscillatory motion include a restoring force that acts to bring the object back toward equilibrium and a tendency for the motion to repeat in time. When the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction, the motion is called simple harmonic motion (SHM).

Basic Terminology in Oscillations

To analyze oscillatory motion, it is essential to understand the following terms:

• Equilibrium Position: The position where the net force on the object is zero.
• Displacement: The distance and direction of the object from the equilibrium position.
• Amplitude: The maximum displacement from equilibrium.
• Period (T): The time required to complete one full oscillation.
• Frequency (f): The number of oscillations per second, given by f = 1/T.
• Angular Frequency (ω): Given by ω = 2πf.

NMAT questions often test your ability to correctly identify and relate these quantities using formulas and graphs.

Simple Harmonic Motion (SHM)

Simple harmonic motion is a special type of oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium and opposite in direction. Mathematically, this is expressed as:

F = −kx

where k is the force constant and x is the displacement. This linear relationship results in sinusoidal motion.

Common examples of SHM include:

• Mass-spring systems
• Simple pendulums (for small angles)
• Vibrations of tuning forks

SHM Equations of Motion

In SHM, displacement, velocity, and acceleration vary with time according to sinusoidal functions:

• Displacement: x = A sin(ωt + φ)
• Velocity: v = Aω cos(ωt + φ)
• Acceleration: a = −Aω² sin(ωt + φ)

Here, A is the amplitude, ω is the angular frequency, t is time, and φ is the phase constant. Note that acceleration is always directed toward the equilibrium position, which is a defining feature of SHM.

Energy in Simple Harmonic Motion

Energy in SHM continuously transforms between kinetic energy (KE) and potential energy (PE). The total mechanical energy remains constant in the absence of damping.

• Kinetic Energy: Maximum at the equilibrium position
• Potential Energy: Maximum at the extreme positions

The total energy of a mass-spring system in SHM is given by:

E = (1/2)kA²

NMAT often includes conceptual questions testing your understanding of energy distribution at different points in the oscillation.

Simple Pendulum

A simple pendulum consists of a mass (bob) suspended by a light, inextensible string. For small angular displacements, the motion of the pendulum approximates SHM.

The period of a simple pendulum is given by:

T = 2π√(L/g)

where L is the length of the pendulum and g is the acceleration due to gravity. Notably, the period does not depend on the mass of the bob or the amplitude (for small angles), a common NMAT concept question.

Damped and Forced Oscillations

In real-world systems, oscillations often lose energy due to friction or air resistance, leading to damped oscillations. In such systems, the amplitude decreases over time.

If an external periodic force is applied to a damped system, it undergoes forced oscillations. When the frequency of the external force matches the natural frequency of the system, resonance occurs, resulting in a large amplitude of oscillation.

Resonance is an important NMAT topic due to its practical implications in engineering, medicine, and structural safety.

Introduction to Waves

A wave is a disturbance that transfers energy from one point to another without the permanent transfer of matter. Waves can be broadly classified into mechanical waves and electromagnetic waves.

• Mechanical Waves: Require a medium (e.g., sound waves, water waves)
• Electromagnetic Waves: Do not require a medium (e.g., light, X-rays)

NMAT Physics focuses primarily on mechanical waves and their properties.

Types of Mechanical Waves

Mechanical waves are classified based on the direction of particle vibration relative to wave propagation:

• Transverse Waves: Particle motion is perpendicular to wave direction (e.g., waves on a string)
• Longitudinal Waves: Particle motion is parallel to wave direction (e.g., sound waves)

Understanding these distinctions is crucial for correctly interpreting wave diagrams and descriptions in NMAT questions.

Wave Parameters and Wave Equation

Important wave parameters include:

• Wavelength (λ): Distance between two consecutive points in phase
• Frequency (f): Number of waves passing a point per second
• Wave Speed (v): Speed at which the wave propagates

These quantities are related by the wave equation:

v = fλ

NMAT frequently tests this fundamental relationship through numerical problems and conceptual questions.

Wave Motion on a String

For waves traveling on a stretched string, the wave speed depends on the tension in the string and its linear mass density:

v = √(T/μ)

where T is the tension and μ is the mass per unit length. Increasing tension increases wave speed, while increasing mass density decreases it.

Sound Waves

Sound waves are longitudinal mechanical waves that propagate through a medium via compressions and rarefactions. The speed of sound depends on the properties of the medium, such as elasticity and density.

Key sound wave characteristics include:

• Pitch: Related to frequency
• Loudness: Related to amplitude or intensity
• Quality (Timbre): Related to waveform

Understanding these relationships helps in answering NMAT questions involving hearing, acoustics, and medical imaging.

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere. This results in fixed points called nodes and points of maximum displacement called antinodes.

Standing waves commonly occur in strings and air columns. The frequencies at which standing waves form are called harmonics.

Doppler Effect

The Doppler effect refers to the apparent change in frequency of a wave due to relative motion between the source and the observer. It is commonly observed in sound waves.

When the source approaches the observer, the observed frequency increases. When the source moves away, the observed frequency decreases. NMAT may test this concept qualitatively or through simplified numerical problems.

Superposition and Interference

The principle of superposition states that when two or more waves overlap, the resulting displacement is the algebraic sum of the individual displacements.

Interference can be constructive or destructive depending on whether the waves reinforce or cancel each other. This principle explains many wave phenomena and is frequently tested in NMAT conceptual questions.

Key NMAT Exam Tips

• Memorize core formulas for SHM and wave motion
• Understand graphs of displacement, velocity, and acceleration
• Focus on conceptual relationships, not just calculations
• Practice identifying wave types and properties from diagrams

Conclusion

Oscillations and waves form a foundational part of NMAT Physics, connecting classical mechanics to real-world applications in sound, light, and medical technology. By mastering the principles of simple harmonic motion, wave behavior, and energy transfer, you can confidently tackle a wide range of NMAT questions. Consistent practice with both conceptual and numerical problems will significantly improve your performance in this topic.

Oscillations and Waves: Problem Sets with Answer Keys (NMAT Physics)

Problem Set 1: Simple Harmonic Motion (Basics)

1. A mass on a spring undergoes SHM with period 0.50 s. What is its frequency?
   1. 0.25 Hz
   2. 0.50 Hz
   3. 2.0 Hz
   4. 3.14 Hz
2. A particle in SHM has amplitude 0.20 m and angular frequency 10 rad/s. What is its maximum speed?
   1. 0.50 m/s
   2. 1.0 m/s
   3. 2.0 m/s
   4. 4.0 m/s
3. In SHM, when is the acceleration magnitude maximum?
   1. At equilibrium
   2. At maximum displacement
   3. At half the amplitude
   4. It is constant
4. A mass-spring system has spring constant k = 200 N/m and mass m = 2.0 kg. What is the period?
   1. 0.20 s
   2. 0.63 s
   3. 1.99 s
   4. 6.28 s
5. A mass in SHM has total energy 8.0 J and amplitude 0.40 m. What is the spring constant?
   1. 25 N/m
   2. 50 N/m
   3. 100 N/m
   4. 200 N/m

Answer Key: Problem Set 1

1. C (f = 1/T = 1/0.50 = 2.0 Hz)
2. C (vmax = Aω = 0.20 × 10 = 2.0 m/s)
3. B (|a| = ω²|x| is maximum at |x| = A)
4. B (T = 2π√(m/k) = 2π√(2/200) ≈ 0.63 s)
5. C (E = 1/2 kA² ⇒ k = 2E/A² = 16/0.16 = 100 N/m)

Problem Set 2: Pendulums, Damping, and Resonance

1. A simple pendulum has length 1.0 m. Take g = 9.8 m/s². What is the period for small oscillations?
   1. 0.64 s
   2. 1.00 s
   3. 2.01 s
   4. 3.14 s
2. If the length of a simple pendulum is quadrupled, the period becomes:
   1. Half
   2. Double
   3. Four times
   4. Unchanged
3. In a damped oscillator, which quantity decreases over time (assuming no external driving force)?
   1. Frequency only
   2. Amplitude only
   3. Period only
   4. Angular frequency increases
4. Resonance occurs when the driving frequency is:
   1. Much smaller than the natural frequency
   2. Much larger than the natural frequency
   3. Equal (or very close) to the natural frequency
   4. Zero
5. A pendulum’s period is independent of:
   1. Length
   2. Gravity
   3. Mass of the bob
   4. All of the above

Answer Key: Problem Set 2

1. C (T = 2π√(L/g) = 2π√(1/9.8) ≈ 2.01 s)
2. B (T ∝ √L, so √4 = 2 → double)
3. B (damping reduces amplitude and energy over time)
4. C (maximum response near natural frequency)
5. C (for small angles, T depends on L and g, not mass)

Problem Set 3: Wave Basics and Wave Speed

1. A wave has frequency 5.0 Hz and wavelength 2.0 m. What is the wave speed?
   1. 2.5 m/s
   2. 7.0 m/s
   3. 10 m/s
   4. 20 m/s
2. If the frequency of a wave doubles while speed stays constant, the wavelength:
   1. Doubles
   2. Halves
   3. Becomes four times
   4. Does not change
3. Which of the following is a longitudinal wave?
   1. Light
   2. Sound in air
   3. Waves on a stretched string
   4. Water surface ripples (idealized as transverse)
4. A string has tension 100 N and linear density 0.020 kg/m. What is the wave speed?
   1. 10 m/s
   2. 50 m/s
   3. 70.7 m/s
   4. 223.6 m/s
5. Increasing the tension in a string (with μ constant) will:
   1. Decrease wave speed
   2. Increase wave speed
   3. Not affect wave speed
   4. Stop wave propagation

Answer Key: Problem Set 3

1. C (v = fλ = 5 × 2 = 10 m/s)
2. B (λ = v/f, so doubling f halves λ)
3. B (sound in air is longitudinal)
4. C (v = √(T/μ) = √(100/0.02) = √5000 ≈ 70.7 m/s)
5. B (v increases as √T)

Problem Set 4: Standing Waves and Harmonics

1. A string fixed at both ends has length 0.80 m. The wave speed is 120 m/s. What is the fundamental frequency?
   1. 37.5 Hz
   2. 75 Hz
   3. 150 Hz
   4. 300 Hz
2. For the same string, what is the frequency of the second harmonic?
   1. 37.5 Hz
   2. 75 Hz
   3. 150 Hz
   4. 300 Hz
3. In a standing wave, a node is a point where:
   1. Displacement is maximum
   2. Displacement is always zero
   3. Energy is maximum
   4. Frequency changes
4. A pipe open at both ends has length 0.50 m. Take speed of sound as 340 m/s. What is the fundamental frequency?
   1. 170 Hz
   2. 340 Hz
   3. 680 Hz
   4. 1360 Hz
5. A pipe closed at one end has length 0.50 m. Take speed of sound as 340 m/s. What is the fundamental frequency?
   1. 85 Hz
   2. 170 Hz
   3. 340 Hz
   4. 680 Hz

Answer Key: Problem Set 4

1. B (fixed-fixed: f1 = v/2L = 120/(2×0.80) = 75 Hz)
2. C (f2 = 2f1 = 150 Hz)
3. B (nodes are zero-displacement points)
4. A (open-open: f1 = v/2L = 340/(2×0.50) = 340 Hz? Wait carefully: 340/1.0 = 340 Hz, so correct choice is B)
5. A (closed-open: f1 = v/4L = 340/(2.0) = 170 Hz, so correct choice is B)

Note: In the two pipe questions above, the correct options are:

• Open-open (L = 0.50 m): f1 = 340 Hz → B
• Closed-open (L = 0.50 m): f1 = 170 Hz → B

Problem Set 5: Interference and Doppler Effect

1. Constructive interference occurs when two waves meet:
   1. In phase
   2. 180° out of phase
   3. With different speeds
   4. Only in a vacuum
2. Destructive interference occurs when the path difference is:
   1. mλ
   2. (m + 1/2)λ
   3. 2mλ
   4. Always zero
3. A sound source approaches a stationary observer. The observed frequency compared to the source frequency is:
   1. Lower
   2. Higher
   3. Unchanged
   4. Zero
4. A stationary source emits sound at 500 Hz. A listener moves toward the source. The observed frequency is:
   1. Less than 500 Hz
   2. Equal to 500 Hz
   3. Greater than 500 Hz
   4. Cannot be determined qualitatively
5. Two speakers emit the same frequency in phase. At a point where the path difference is exactly one wavelength, the interference is:
   1. Destructive
   2. Constructive
   3. Neither
   4. Random

Answer Key: Problem Set 5

1. A
2. B
3. B
4. C
5. B (path difference = 1λ = mλ → constructive)

Bonus: Mixed NMAT-Style Short Problems (Computation)

1. A mass-spring system has m = 0.50 kg and k = 80 N/m. Find the frequency.
2. A wave travels with speed 24 m/s and frequency 8 Hz. Find the wavelength.
3. A pendulum’s period is 1.8 s on Earth. If taken to a planet where g is 4 times Earth’s, what is the new period?
4. A 0.60 m string fixed at both ends has wave speed 90 m/s. Find the third harmonic frequency.
5. In SHM, if amplitude doubles while ω stays constant, what happens to the total energy?

Answer Key: Bonus (with final answers)

1. f ≈ 2.01 Hz (ω = √(k/m) = √(80/0.5)=√160≈12.65 rad/s, f = ω/2π≈2.01)
2. λ = 3.0 m (λ = v/f = 24/8)
3. T = 0.90 s (T ∝ 1/√g, so g×4 → T/2)
4. f3 = 225 Hz (f1 = v/2L = 90/(1.2)=75 Hz, f3=3f1=225 Hz)
5. Energy becomes 4 times (E ∝ A²)

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