Vibrations

AS UNIT 3

Oscillations and Nuclei

2 Vibrations 

Learners should be able to demonstrate and apply their knowledge and understanding of:
a) The definition of simple harmonic motion as a statement in words
b)  [math]a = -ω^2 x[/math] as a mathematical defining equation of simple harmonic motion
c) The graphical representation of the variation of acceleration with displacement during simple harmonic motion
d) [math]x = Acos(ωt + ε)[/math] as a solution to [math]a = -ω^2 x[/math]
e) The terms frequency, period, amplitude and phase
f) Period as [math] \frac{1}{f}  \text{or} \frac{2π}{ω}[/math]
g) [math]v = -Aω sin⁡(ωt + ε)[/math] for the velocity during simple harmonic motion
h) The graphical representation of the changes in displacement and velocity with time during simple harmonic motion
i) The equation [math]T = 2\pi \sqrt{\frac{m}{k}}[/math] for the period of a system having stiffness (force per unit extension) k and mass m
j) The equation [math]T = 2\pi \sqrt{\frac{l}{g}}[/math] for the period of a simple pendulum
k) The graphical representation of the interchange between kinetic energy and potential energy during undamped simple harmonic motion, and perform simple calculations on energy changes
l) Free oscillations and the effect of damping in real systems
m) Practical examples of damped oscillations
n) The importance of critical damping in appropriate cases such as vehicle suspensions
o) Forced oscillations and resonance, and to describe practical examples
p) The variation of the amplitude of a forced oscillation with driving frequency and that increased damping broadens the resonance curve
q) Circumstances when resonance is useful for example, circuit tuning, microwave cooking and other circumstances in which it should be avoided for example, bridge design

Specified Practical Work

o   Measurement of g with a pendulum

o   Investigation of the damping of a spring

  • a)   Definition in Words

  • Simple Harmonic Motion (SHM) is the oscillatory motion of an object about an equilibrium position where the restoring force is directly proportional to the displacement from that equilibrium and always acts in the opposite direction.
  • Figure 1 Simple harmonic motion
  • In simpler terms, if you pull or push an object away from its rest position, it will experience a force that tries to bring it back, and this force increases the further the object is displaced.

  • b)   Mathematical Defining Equation: [math]a = -ω^2 x[/math]

  • ⇒ Understanding the Equation
  • Acceleration (a):
  • – In SHM, the acceleration of the object is not constant; instead, it changes in such a way that it is always directed opposite to the displacement.
  • Displacement (x):
  • – This is the distance of the object from its equilibrium (or rest) position. Positive x indicates displacement in one direction, and negative x indicates displacement in the opposite direction.
  • Angular Frequency (ω):
  • – This is a measure of how rapidly the oscillations occur. It is related to the period T (the time for one complete cycle) by:
  • [math]ω = \frac{2π}{T}[/math]
  • The Equation [math]a = -ω^2 x:[/math] :
  • The negative sign indicates that the acceleration is always directed opposite to the displacement. For example, if x is positive (object is displaced to the right of equilibrium), then a is negative (acceleration is directed to the left, toward equilibrium).
  • The magnitude of the acceleration is proportional to the displacement: the further the object is from equilibrium, the greater the acceleration that will act to restore it.
  • ⇒  Physical Implication
  • This equation encapsulates the essence of SHM: it is the mathematical statement that describes how an object’s acceleration is directly proportional to its displacement but oppositely directed. Many systems in physics, such as a mass on a spring or a simple pendulum (for small angles), follow this behavior.

  • c)    Graphical Representation: Acceleration vs. Displacement

  • When we graph the acceleration a against displacement x for an object in SHM, the resulting graph is a straight line that passes through the origin with a negative slope.
  • Figure 2 Graphically representation
  • ⇒  Graph Characteristics
  • – X-Axis: Displacement x.
  • – Y-Axis: Acceleration a.
  • – Line Equation:
  • From [math]a = -ω^2 x:[/math]:, we see that the line has a slope of .
  • ⇒  Interpretation of the Graph
  • Linear Relationship: The straight-line graph shows that the acceleration is directly proportional to the displacement.
  • Negative Slope: The negative value of the slope confirms that the acceleration is always opposite in direction to the displacement.
  • Zero at Equilibrium: When x=0 (the equilibrium position), the acceleration a is zero. This is consistent with the fact that at the equilibrium point, no net restoring force is acting on the object.
  • This graphical representation provides an immediate visual understanding of the restoring nature of the force in SHM.

  • d)    General Solution: [math]x = Acos(ωt + ε)[/math]

  • The differential equation for SHM:
  • [math]a = \frac{d^2 x}{dt^2} \\
    a = -\omega^2 x[/math]
  • has a general solution given by:
  • [math]x = Acos(ωt + ε)[/math]
  • Where:
  • – A (Amplitude): The maximum displacement from the equilibrium position. It represents the “size” of the oscillation.
  • – ω (Angular Frequency): As described above, it determines how fast the oscillation occurs.
  • – t (Time): The independent variable representing time.
  • – ϵ (Phase Constant): This constant accounts for the initial conditions of the motion (i.e., the value of x and v at t=0).
  • ⇒  Understanding the Solution
  • Cosine Function: The use of the cosine function indicates that the motion is periodic and oscillatory.
  • Periodic Behavior: Since the cosine function is periodic with period 2π, the motion repeats every time t increases by ​[math]T = \frac{2π}{ω}.[/math].
  • Phase Shift ϵ\epsilonϵ: Depending on the initial displacement and velocity of the oscillating object, the phase constant ϵ\epsilonϵ shifts the cosine curve horizontally, ensuring that the solution fits the initial conditions of the problem.
  • ⇒  Example
  • Suppose an object in SHM has:
  • – Amplitude A = 5cm,
  • – Angular frequency ω = 2 rad/s
    – Phase constant ϵ = 0.
  • Then the position as a function of time is:
  • [math]x(t) = 5 cos(2t)[/math]
  • This equation tells us that the object oscillates between +5 cm and −5 cm, completing one full cycle every [math]T = \frac{2π}{2} = πs[/math].
  • Below is a detailed explanation of these fundamental concepts in simple harmonic motion (SHM), including definitions of key terms, the relationships between period, frequency, amplitude, phase, and the expressions for displacement and velocity, along with descriptions of their graphical representations.

  • e)    Key Terms

  • ⇒  Frequency (f)
  • Definition:
  • – Frequency is the number of oscillations (cycles) per unit time. It tells us how many complete cycles of the motion occur in one second.
  • Units:
  • – Measured in hertz (Hz), where 1Hz = 1cycle per second
    ⇒   Period (T)
  • Definition:
  • – The period is the time taken for one complete cycle of the oscillatory motion.
  • Relationship with Frequency:
  • [math]T = \frac{1}{f}[/math]
  • Alternatively, when the motion is described in terms of angular frequency ω, the period is:
  • [math]T = \frac{2π}{ω}[/math]
  • Units:
  • – Measured in seconds (s).
  • Figure 3 Basic terms of a wave
  • ⇒   Amplitude (A)
  • Definition:
  • – Amplitude is the maximum displacement of the oscillating object from its equilibrium (or rest) position. It represents the “size” or “strength” of the oscillation.
  • Units:
  • – The same units as displacement (e.g., meters, centimeters).
  • ⇒   Phase (ϵ)
  • Definition:
  • – The phase (or phase constant) determines the initial condition of the motion; it is the horizontal shift of the oscillatory function. The phase tells us the state (position and velocity) of the oscillator at time t=0.
  • Units:
  • – Measured in radians (or degrees).

  • f) Period As [math]\frac{1}{f} \text{or} \frac{2π}{ω}[/math]

  • The period of a periodic motion is the time it takes for one complete cycle. Two common ways to express the period T are in terms of the frequency f or the angular frequency ω.
  • 1. Using Frequency f
  • Frequency is defined as the number of cycles (oscillations) per second.
  • Since the period is the time for one cycle, it is simply the reciprocal of the frequency:
  • [math]T = \frac{1}{f}[/math]
  • For example, if an oscillator has a frequency of 5 Hz (meaning 5 cycles per second), then the period is:
  • [math]T = \frac{1}{f} \\
    T = \frac{1}{5} \text{ seconds} \\
    T = 0.2 \text{ seconds}[/math]
  • 2. Using Angular Frequency ω
  • Angular frequency is related to the regular frequency by:
  • [math]ω = 2πf[/math]
  • Here, ω is measured in radians per second.
  • Since ​[math]f = \frac{ω}{2π}[/math], substituting this into ​[math]T = \frac{1}{f}[/math] gives:
  • [math]T = \frac{1}{\left(\frac{\omega}{2\pi}\right)} \\
    T = \frac{2\pi}{\omega}[/math]
  • Reciprocal Relationship with Frequency:
  • – Frequency counts how many cycles occur per unit time. If a process occurs f times per second, then each cycle takes  [math]\frac{1}{f}[/math]​
  • Relation Between f and ω:
  • – The factor 2π comes in because a complete cycle corresponds to a full circle (360° or 2π radians). Thus, the angular frequency tells us how many radians are swept out per second. Dividing the full circle 2π by the rate ω (in radians per second) gives the time for one full cycle.

  • g)   Mathematical Expressions in Simple Harmonic Motion

  • ⇒   Displacement Equation
  • The displacement of an object undergoing SHM as a function of time is given by:
  • [math]x(t) = Acos(ωt + ϵ)[/math]
  • – x(t) is the displacement at time t.
  • – A is the amplitude.
  • – ω is the angular frequency, related to the period by ​[math]ω = \frac{2π}{T}[/math].
  • – ϵ is the phase constant.
  • ⇒   Velocity Equation
  • The velocity of the object is the first-time derivative of the displacement:
  • [math]v(t) = \frac{dx}{dt} \\
    v(t) = -A\omega \sin(\omega t + \epsilon)[/math]
  • This expression shows that the velocity is sinusoidal, with the same angular frequency ω and phase constant ϵ as the displacement.
  • The negative sign indicates that the velocity is maximum (in magnitude) when the displacement is zero, and it is zero when the displacement is at a maximum (or minimum).

  • h)   Graphical Representations

  • ⇒  Displacement vs. Time
  • Graph Shape:
  • – A cosine wave (or sine wave if the phase constant is adjusted) oscillates between +A and −A.
  • Characteristics:
  • – Amplitude: The highest and lowest points of the graph are at +A and −A, respectively
  • – Period: The distance along the time-axis for one complete cycle is T (or ​[math]\frac{2π}{ω}[/math]).
  • – Phase Shift: The graph is shifted horizontally by an amount determined by ϵ\epsilonϵ. For example, if [math]ϵ = 0[/math], the cosine graph starts at its maximum value at t=0.
  • – Interpretation:
    The cosine graph shows that the displacement varies periodically with time, returning to the same value every period T.
  • Figure 4 Graphically representation between time and displacement
  • ⇒   Velocity vs. Time:
  • Graph Shape:
  • – A sine wave (multiplied by – Aω) that oscillates between + Aω and – Aω.
  • Figure 5 Graphically representation between velocity and time
  • Characteristics:
  • – Amplitude: The maximum speed is Aω
  • Phase Relationship:
  • – The velocity graph is 90° (or π/2 radians) out of phase with the displacement graph.
  • – When the displacement is zero (crossing the equilibrium position), the velocity is at a maximum (in magnitude).
  • – When the displacement is at a maximum (or minimum), the velocity is zero.
  • Interpretation:
    The sine wave for velocity indicates that the speed reaches its maximum as the object passes through the equilibrium position and decreases to zero at the turning points (maximum displacement).
  • ⇒   Combined Graph Interpretation
  • Displacement and Velocity Relationship:
  • – A combined diagram might show:
  • – A cosine curve for displacement x(t).
  • – A sine curve (with an appropriate negative sign) for velocity v(t)v(t)v(t).
  • Visual Comparison:
  • – At t = 0, if ϵ = 0, x(0) = A (maximum displacement) and v(0) = 0
  • – As time increases, the displacement decreases towards zero, while the velocity increases in magnitude, reaching its maximum when x = 0.

  • i)   Period of a Mass-Spring System

  • ⇒  Formula:
  • [math]T = 2\pi \sqrt{\frac{m}{k}}[/math]
  • ⇒ Explanation:
  • m is the mass attached to the spring.
  • k is the spring constant (stiffness), defined as the force required per unit extension ([math]N/m[/math]).
  • The formula is derived from Newton’s second law applied to a spring:
  • – For a mass-spring system, the restoring force is given by Hooke’s law:
  • [math]F = -kx[/math]
  • Figure 6 Mass attach to spring system
  • – Applying Newton’s law ([math]F = ma[/math] ) and writing acceleration as the second derivative of displacement, we have:
  • [math]m \frac{d^2 x}{dt^2} = -kx[/math]
  • – This leads to the differential equation for SHM:
  • [math]\frac{d^2 x}{dt^2} + \frac{k}{m} x = 0[/math]
  • – The general solution is a sinusoidal function with angular frequency:
  • [math]\omega = \sqrt{\frac{k}{m}}[/math]
  • – The period T is then the time for one full cycle:
  • [math]T = \frac{2\pi}{\omega} \\
    T = 2\pi \sqrt{\frac{m}{k}}[/math]
  • ⇒  Interpretation:
  • A larger mass mmm increases the period (slower oscillations).
  • A stiffer spring (larger k) decreases the period (faster oscillations).
  • ⇒ A Mass on a spring:
  • In a mass-spring system, the kinetic energy is given by [math]\frac{1}{2} k x^2[/math] potential energy is called elastic potential energy and it is stored in the spring when extended. When the spring is extended by x, the elastic potential energy is [math]\frac{1}{2} k x^2[/math] where k is the stiffness.
  • Because the total energy must be the same we can say:
  • [math]\frac{1}{2} k A^2 = \frac{1}{2} m v^2 + \frac{1}{2} k x^2 \\
    v^2 = \frac{k}{m} (A^2 – x^2) \\
    v = \pm \omega \sqrt{A^2 – x^2}[/math]

  • j)   Period of a Simple Pendulum

  • ⇒  Formula:
  • [math]T = 2\pi \sqrt{\frac{l}{g}}[/math]
  • ⇒  Explanation:
  • – l is the length of the pendulum.
  • – g is the acceleration due to gravity.
  • – For small angular displacements (small-angle approximation where  [math]sinθ ≈ θ[/math] in radians), the restoring force on a pendulum bob leads to a differential equation analogous to that of a mass-spring system:
  • – The restoring acceleration is proportional to the displacement angle.
  • – The differential equation becomes:
  • [math]\frac{d^2 x}{dt^2} + \frac{k}{m} x = 0[/math]
  • Its solution is sinusoidal with angular frequency:
  • [math]\omega = \sqrt{\frac{g}{l}}[/math]
  • Thus, the period is:
  • [math]T = \frac{2\pi}{\omega} \\
    T = 2\pi \sqrt{\frac{l}{g}}[/math]
  • Figure 7 Simple pendulum
  • ⇒  Interpretation:
  • A longer pendulum (larger l) results in a longer period.
  • A stronger gravitational field (larger g) leads to a shorter period.
  • ⇒ Pendulum:
  • In a pendulum system, we have gravitational potential energy instead of elastic potential energy.
  • [math]L – L \cos(\theta_{\max}) = h_{\max} \\
    m g h_{\max} = \frac{1}{2} m v^2 + m g h \\
    g h_{\max} = \frac{v^2}{2} + g h \\
    v^2 = 2g (h_{\max} – h) = 2gL \left( (1 – \cos(\theta_{\max})) – (1 – \cos\theta) \right) \\
    v^2 = 2gL (\cos\theta – \cos\theta_{\max})[/math]
  • Note, you do not need to know these equations but they may be helpful because you may have to work out the speed of the pendulum bob at a certain angle.

  • k)    Energy Interchange in Undamped Simple Harmonic Motion

  • ⇒  Total Energy in SHM:
  • For an undamped oscillator (no energy loss), the total mechanical energy [math][/math] remains constant and is the sum of kinetic and potential energy.
  • Potential Energy in a Spring:
  • [math]U = \frac{1}{2} k x^2[/math]
  • Kinetic Energy:
  • [math]K = \frac{1}{2} m v^2[/math]
  • Total Energy:
  • [math]E_{\text{total}} = \frac{1}{2} m v^2 + \frac{1}{2} k x^2[/math]
  • For SHM, [math]E_\text{{total}}[/math]​ is constant and equals [math]\frac{1}{2} k A^2[/math] where A is the amplitude.
  • ⇒  Graphical Representation:
  • 1. Displacement vs. Time:
  • – A cosine (or sine) wave with amplitude A and period T.
  • – Example: [math]x(t) = Acos(ωt + ϵ)[/math]
  • 2. Velocity vs. Time:
  • – A sine (or cosine with a phase shift) wave representing [math]v(t) = -Aω sin⁡(ωt + ϵ)[/math]
  • – Note the 90° phase difference: when displacement is maximum, velocity is zero; when displacement crosses zero, velocity is maximum.
  • 3. Energy vs. Time:
  • Potential Energy (PE): Varies as
  • [math]U = \frac{1}{2} k x (t)^2[/math]
  • – It is maximum when [math][/math](turning points) and zero at the equilibrium position.
  • Kinetic Energy (KE): Varies as
  • [math]K(t) = \frac{1}{2} mv(t)^2[/math]
  • – It is maximum at the equilibrium position (when x=0) and zero at the turning points
  • – When plotted, KE and PE vary inversely, but their sum remains constant. The graphs are typically sinusoidal functions (or squared sinusoids), with one peaking when the other is at a minimum.
  • ⇒  Simple Calculation Example:
  • Consider a mass-spring system with:
  • m = 0.5 kg,
  • k = 200 N/m,
  • Amplitude A = 0.1 m.
  • 1. Period Calculation:
  • [math]T = 2\pi \sqrt{\frac{m}{k}} \\
    T = 2\pi \sqrt{\frac{0.5}{200}} \\
    T = 2\pi \sqrt{0.0025} \\
    T = 2\pi (0.05) \\
    T = 0.314 \text{ s}[/math]
  • 2. Total Energy:
  • [math]E_{\text{total}} = \frac{1}{2} k A^2 \\
    E_{\text{total}} = \frac{1}{2} (200) (0.1)^2 \\
    E_{\text{total}} = 1 \text{ Joule}[/math]
  • 3. At a Displacement x=0.05 m:
  • – Potential Energy:
  • [math]U = \frac{1}{2} k x(t)^2 \\
    U = \frac{1}{2} (200) (0.05)^2 \\
    U = 0.25 \text{ Joule}[/math]
  • – Kinetic Energy (since [math][/math]) :
  • [math]K = E_{\text{total}} – U \\
    K = 1 – 0.25 \\
    K = 0.75 \text{ Joule}[/math]

  • l. Free Oscillations and the Effect of Damping in Real Systems

  • ⇒  Free Oscillations:
  • Definition:
  • – Free oscillations occur when a system is displaced from its equilibrium position and allowed to oscillate without any external driving force.
  • Undamped Oscillations:
  • – In an ideal system with no friction or other forms of energy loss, the amplitude remains constant and the motion is perfectly sinusoidal.
  • ⇒  Damped Oscillations:
  • In real systems, damping (e.g., friction, air resistance) causes energy to be lost over time, leading to a gradual decrease in the amplitude of oscillation. The equation of motion for a damped oscillator is:
  • [math]m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = 0[/math]
  • Where:
  • – b is the damping coefficient.
  • Figure 8 Damped oscillation
  • ⇒  Effects of Damping:
  • 1. Exponential Decay of Amplitude:
  • – The amplitude decreases as:
  • [math]A(t) = A_0 e^{-\frac{k}{2m} t}[/math]
  • – Where ​[math]A_o[/math] is the initial amplitude.
  • 2. Shift in Frequency:
  • – The damped angular frequency is given by:
  • [math]\omega_d = \sqrt{\frac{k}{m} – \left(\frac{b}{2m}\right)^2}[/math]
  • This is slightly lower than the undamped frequency
  • [math]\omega = \sqrt{\frac{k}{m}} \\
    \omega^2 = \frac{k}{m}[/math]
  • 3. Graphical Representation:
  • ⇒ Undamped Oscillations:
  • – Displacement vs. time shows a sine or cosine wave with constant amplitude.
  • ⇒ Damped Oscillations:
  • – Displacement vs. time shows a sine or cosine wave whose envelope decays exponentially over time. Energy graphs show the total mechanical energy decreasing exponentially.
  • ⇒  Example of Damped Oscillation:
  • For a system with m=1 kg, k=100 N/m, and damping coefficient b=2kg/s:
  • Undamped angular frequency:
  • [math]\omega = \sqrt{\frac{100}{1}} \\
    \omega = 10 \text{ rad/s}[/math]
  • Damped angular frequency:
  • [math]\omega_d = \sqrt{\frac{k}{m} – \left(\frac{b}{2m}\right)^2} \\
    \omega_d = \sqrt{\omega^2 – \left(\frac{b}{2m}\right)^2} \\
    \omega_d = \sqrt{(10)^2 – \left(\frac{2}{2}\right)^2} \\
    \omega_d = \sqrt{100 – 1} \\
    \omega_d = \sqrt{99} \\
    \omega_d = 9.95 \text{ rad/s}[/math]
  • The amplitude decays as:
  • [math]A(t) = A_0 e^{-t/1}[/math]
  •    Here, [math]\frac{b}{2m} = \frac{2}{2} = 1s^{-1}[/math]

  • m) Practical Examples of Damped Oscillations

  • Damped oscillations occur when an oscillating system loses energy over time (for example, through friction, air resistance, or other dissipative forces). In many real-world systems, we do not see ideal undamped motion; instead, the amplitude of oscillations gradually decreases. Some practical examples include:
  • 1. Swinging Pendulum in Air:
  • A pendulum (such as in a grandfather clock) eventually comes to rest because air resistance and friction at the pivot dissipate energy.
  • 2. Mass-Spring Systems:
  • A weight attached to a spring (like in a shock absorber or a laboratory mass-spring setup) will oscillate with a gradually decreasing amplitude due to internal friction in the spring and air drag.
  • 3. Vibrations in Mechanical Structures:
  • Buildings or bridges experience oscillations (from wind or earthquakes) that are damped by structural damping, converting vibrational energy into heat.
  • 4. Electronic Oscillators:
  • In circuits (e.g., RLC circuits), resistance causes the oscillatory voltage or current to decay over time.
  • 5. Car Suspension:
  • Although discussed more in critical damping below, even ordinary suspension systems show damped oscillations after hitting a bump; the shock absorbers dissipate energy to prevent continual bouncing.

  • n) The Importance of Critical Damping in Appropriate Cases (e.g., Vehicle Suspensions)

  • Critical damping is a specific amount of damping that brings an oscillating system to rest as quickly as possible without oscillating. It is the ideal case between underdamping (where the system oscillates while decaying) and overdamping (where the system returns to equilibrium slowly without oscillation).
  • ⇒   Critical Damping Matters:
  • 1. Rapid Stabilization:
  • In systems such as vehicle suspensions, you want the system (the car body) to return to equilibrium quickly after a disturbance (like hitting a bump) without overshooting or oscillating. Critical damping provides the fastest return to equilibrium without any additional bounces.
  • 2. Comfort and Safety:
  • In a vehicle, excessive oscillations (underdamping) can lead to a bouncy ride and loss of control, while too much damping (overdamping) can make the ride harsh and less responsive.
  • Critical damping offers an optimum balance, smoothing out shocks while maintaining handling.
  • 3. Efficiency in Mechanical Systems:
  • Critical damping minimizes energy loss while preventing sustained oscillations, which is important in precision instruments, door closers, and even in some aerospace applications.
  • ⇒  Vehicle Suspension Example:
  • Shock Absorbers:
  • – Modern car suspensions are designed to approximate critical damping so that after a bump, the car quickly settles to a smooth ride without prolonged bouncing. This not only improves passenger comfort but also enhances road safety by maintaining better tire contact with the road.

  • o) Forced Oscillations and Resonance, with Practical Examples

  • Forced oscillations occur when an external periodic force drives a system. Unlike free oscillations, which occur with an initial displacement and then decay over time, forced oscillations are maintained by the continuous application of energy.
  • 1. Driving Force:
  • The system is subjected to an external force that varies periodically
  • (e.g.,[math]F(t) = F_0 cos⁡(ω_{drive} t)[/math])
  • 2. Resonance:
  • Resonance occurs when the driving frequency ​[math]ω_{drive}[/math] matches the natural frequency ​[math][/math] of the system.
  • At resonance, the amplitude of the oscillations can become very large because the energy from the driving force is added in phase with the motion.
  • 3. Damping and Resonance:
  • In a damped system, the maximum amplitude at resonance is lower than in an undamped system, but the concept of resonance still applies.
  • Excessive amplitude at resonance can cause damage (e.g., in buildings during earthquakes).
  • ⇒  Practical Examples:
  • 1. Musical Instruments:
  • A guitar string vibrates with large amplitude when played at its resonant frequency. The body of the instrument further resonates, amplifying sound.
  • 2. Bridges and Buildings:
  • The collapse of the Tacoma Narrows Bridge in 1940 is a classic (if tragic) example of resonance. Wind-induced vibrations at a frequency matching the bridge’s natural frequency caused large amplitude oscillations and structural failure.
  • 3. Microwave Ovens:
  • The cavity in a microwave oven is designed to resonate with microwaves, leading to efficient energy transfer and heating of food.
  • 4. Electrical Circuits:
  • RLC circuits can exhibit resonance when driven at a frequency that matches the circuit’s natural frequency. This is used in filters and tuning circuits in radios.
  • 5. Mechanical Clocks and Watches:
  • The balance wheel in a mechanical watch is subject to forced oscillations (through the escapement mechanism) and is designed to oscillate at a precise resonant frequency for accurate timekeeping.
  • ⇒  Graphical Representation:
  • Amplitude vs. Driving Frequency Curve:
  • – When plotting the amplitude of the forced oscillation against the driving frequency, a sharp peak is observed at the resonant frequency, indicating maximum energy transfer.
  • – The width of the resonance peak depends on the damping: less damping yields a sharper, higher peak; more damping produces a broader, lower peak.

  • p. Forced Oscillations and Resonance: Amplitude Variation with Driving Frequency

  • When a system is subjected to an external periodic driving force, it undergoes forced oscillations. The amplitude of these oscillations depends on the driving frequency and the system’s natural frequency. The relationship between amplitude and frequency can be visualized using a resonance curve (also known as a frequency response curve).
  • ⇒  Resonance and Amplitude Response
  • When the driving frequency fd​ is far from the natural frequency f0​, the system oscillates with a relatively low amplitude.
  • As fd approaches f0​, the amplitude increases significantly.
  • At resonance fd = f0​), the system reaches maximum amplitude.
  • Beyond this point (fd> f0​), the amplitude starts to decrease.
  • The equation governing the amplitude A of forced oscillations is:
  • [math]A = \frac{F_0 / m}{\sqrt{(\omega_0^2 – \omega^2)^2 + (2\beta\omega)^2}}[/math]
  • Where:
  • [math]F_o[/math] is the driving force amplitude,
  • m is the mass of the system,
  • [math]\omega_o[/math] is the natural angular frequency,
  • [math]\omega[/math] is the driving angular frequency,
  • [math]\beta[/math] is the damping coefficient.
  • ⇒  Effect of Damping on the Resonance Curve
  • Damping affects the sharpness and height of the resonance peak:
  • 1. Low Damping (Under-damped System):
  • – Sharp and high resonance peak.
  • – The system oscillates with very high amplitude near ​[math]f_o[/math].
  • – Example: A lightly damped tuning fork rings loudly when struck.
  • 2. Increased Damping:
  • – The peak lowers and broadens, meaning the system respond to a wider range of frequencies with lower amplitude.
  • – Example: A car’s suspension system is designed with damping to prevent excessive oscillations when driving over bumps.
  • 3. Very High Damping (Over-damped System):
  • – No clear resonance peak, and the system does not oscillate significantly.
  • – Example: Shock absorbers in vehicles, where excessive oscillations are undesirable.
  • The quality factor Q describes how sharp the resonance is:
  • [math]Q = \frac{f_0}{\Delta f}[/math]
  • Where [math][/math] is the frequency range over which the amplitude is significantly large. High Q means sharp resonance, while low Q means broad resonance.

  • q. Practical Applications of Resonance

  • Useful Applications of Resonance
  • 1. Circuit Tuning (Radio and TV Reception)
  • Resonance is used in LC circuits (inductor-capacitor circuits) to select specific frequencies from a broad range of signals.
  • The circuit resonates at the frequency of the desired radio or TV signal while rejecting others.
  • 2. Microwave Cooking
  • Microwaves operate at 45 GHz, the resonant frequency of water molecules.
  • The energy is absorbed efficiently by water in food, causing heating.
  • 3.Musical Instruments
  • String and wind instruments rely on resonance to amplify sound at particular frequencies.
  • Example: A violin’s body resonates at specific frequencies, enhancing the sound quality.
  • 4. MRI (Magnetic Resonance Imaging)
  • Resonance of atomic nuclei in a magnetic field helps create detailed medical images.
  • 5. Quartz Watches
  • Quartz crystals vibrate at a precise resonance frequency, allowing accurate timekeeping.
  • ⇒  When Resonance Should Be Avoided
  • 1. Bridge and Building Design
  • – If the natural frequency of a bridge or building matches external vibrations (wind, earthquakes, or pedestrians), resonance can cause destructive oscillations.
  • – Example: The Tacoma Narrows Bridge collapse (1940) due to wind-induced resonance.
  • – Modern buildings use tuned mass dampers to counteract resonance.
  • 2. Machinery and Vehicles
  • – Rotating parts in engines, turbines, and aircraft must avoid resonance frequencies to prevent excessive vibrations and damage.
  • – Example: Aircraft wings and helicopter blades are tested to prevent resonance-related failures.
  • 3. Earthquake Engineering
  • – Buildings are designed with shock absorbers or flexible foundations to prevent resonance with seismic waves.
  • 4. Glass and Structures
  • – Certain frequencies can cause glass shattering due to resonance (e.g., a singer breaking a wine glass with their voice).
  • ⇒   Conclusion
  • Resonance is a powerful phenomenon that can be both beneficial and destructive. Understanding and controlling resonance is crucial in fields ranging from engineering and medicine to music and daily appliances. While it enables efficient energy transfer in useful applications, excessive resonance must be mitigated to prevent catastrophic failures in structures and machinery.

  • Specified Practical Work

  • Experiment 1: Measurement of g with a Simple Pendulum

  • ⇒ Objective:
  • To determine the acceleration due to gravity (g) by measuring the period of a simple pendulum.
  • ⇒  Apparatus:
  • Retort stand with a clamp
  • String (approximately 1.5m)
  • Small dense bob (e.g., metal or wooden sphere)
  • Stopwatch (accuracy of at least 0.01s)
  • Meter ruler
  • Protractor
  • Figure 9  Measurement of g with a simple pendulum
  • ⇒   Theory:
  • For a simple pendulum of length L, the time period T is given by:
  • [math]T = 2\pi \sqrt{\frac{L}{g}}[/math]
  • Squaring both sides:
  • [math]T^2 = \frac{4\pi^2 L}{g}[/math]
  • By plotting [math]T^2[/math] against L, we can determine g from the slope.
  • ⇒   Procedure:
  • 1. Setup the Pendulum:
  • – Fix one end of the string to a clamp stand so that it hangs freely.
  • – Attach a small bob to the free end of the string.
  • – Measure the length of the pendulum (L) from the fixed point to the center of the bob using a meter ruler.
  • 2. Oscillation Measurement:
  • – Displace the bob slightly (small angle [math]10^o[/math]) to ensure simple harmonic motion.
  • – Release the bob gently without applying any external force.
  • – Use the stopwatch to measure the time for 10 complete oscillations.
  • – Record the time and divide by 10 to get the time period T.
  • 3. Repeat the Measurements:
  • – Vary the length L (e.g., 30 cm, 40 cm, 50 cm, 60 cm, 70 cm, 80 cm, etc.).
  • – For each length, repeat the time measurement at least three times and take an average.
  • ⇒   Data Analysis:
  • Calculate [math]T^2[/math] for each length.
  • Plot a graph of [math]T^2[/math](y-axis) against L (x-axis).
  • Determine the slope mmm of the best-fit line.
  • Since the equation is:
  • [math]T^2 = \frac{4\pi^2 L}{g}[/math]
  • The acceleration due to gravity is:
  • [math]g = \frac{4\pi^2}{\text{slope}}[/math]
  • ⇒   Precautions:
  • Use a small amplitude to avoid deviations from simple harmonic motion.
  • Ensure the string is inextensible and has negligible mass.
  • Start the stopwatch at the correct moment to minimize human error.
  • Avoid air resistance by using a dense bob.
  • Experiment 2: Investigation of the Damping of a Spring
  • ⇒  Objective:
  • To study the damping effect in an oscillating spring system.
  • ⇒   Apparatus:
  • Clamp stand with a spring holder
  • Helical spring
  • Masses (e.g., 50g, 100g, 150g, etc.)
  • Meter ruler
  • Stopwatch
  • Damping material (water, oil, foam for variations)
  • ⇒   Theory:
  • When a mass is attached to a vertical spring and set into oscillation, the amplitude decreases over time due to damping forces like air resistance and internal friction. The displacement x follows:
  • [math]x = x_0 e^{-\gamma t} \cos(\omega t + \varphi)[/math]
  • Where γ is the damping coefficient.
  • By measuring the amplitude over time, we can determine the effect of damping.
  • Figure 10 Investigation of the damping of a spring
  • ⇒   Procedure:
  • 1. Setup the System:
  • – Suspend the spring from the clamp stand.
  • – Attach a known mass to the free end.
  • – Measure the equilibrium position using a meter ruler.
  • 2. Oscillation Measurement:
  • – Pull the mass down slightly and release it without applying extra force.
  • – Start the stopwatch and measure the amplitude of oscillation after each complete cycle.
  • – Repeat the measurement for multiple cycles.
  • 3. Introduce Damping:
  • – Repeat the experiment by introducing different damping methods:
  • – Submerging the mass partially in water.
  • – Wrapping the spring with foam.
  • – Using a high-viscosity fluid (e.g., oil).
  • 4. Record the Data:
  • – Note the amplitude x at different times.
  • – Plot [math]Inx[/math] against time t to find the damping coefficient γ.
  • ⇒  Data Analysis:
  • The slope of the [math]Inx[/math] vs. t graph gives γ.
  • Compare the effect of different damping conditions.
  • ⇒   Precautions:
  • Ensure the mass moves vertically without lateral motion.
  • Minimize external disturbances (e.g., wind or vibration).
  • Use a stopwatch with high precision.
  • Repeat measurements for accuracy.
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