Mechanics & Oscillations: Forces, Motion, Springs, Pendulums & Waves

TAKEAWAYS:

• Newton’s laws explain motion: inertia, force = mass × acceleration, and action-reaction pairs govern everything from falling objects to rocket launches.
• Hooke’s law ($F=-kx$) describes springs and elastic materials, critical for designing suspension systems (e.g., cars) and musical instruments (e.g., guitars).
• Simple harmonic motion (SHM) is periodic motion where restoring force is proportional to displacement (e.g., pendulums, springs, sound waves).
• Energy in SHM alternates between kinetic and potential, with total energy conserved—key for analyzing oscillators like quartz watches or seismic sensors.
• Waves transmit energy via oscillations (transverse/longitudinal), explaining phenomena from light to sound and enabling technologies like fiber optics and ultrasound.
• Damped oscillations (real-world systems lose energy to friction/air resistance) are modeled mathematically to predict how systems like car shocks or building foundations behave over time.

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1. Kinematics: Describing Motion

Motion is the change in position of an object over time. Kinematics studies this without considering forces.

Key Definitions:

• Displacement (s): Change in position (vector quantity, SI unit: meters).
• Velocity (v): Rate of change of displacement ($v=\frac{ds}{dt}$), includes direction.
• Acceleration (a): Rate of change of velocity ($a=\frac{dv}{dt}$), caused by forces.
• Equations of Motion (for constant acceleration): $$v=u+at,s=ut+\frac{1}{2}a{t}^2,v^2=u^2+2as$$ where $u$ = initial velocity, $a$ = acceleration, $t$ = time.

Worked Example: Ncell Delivery Bike

A Pathao delivery bike accelerates uniformly from rest to 15 m/s in 5 seconds. Calculate:

1. Its acceleration.
2. Distance covered in this time.

Solution:

1. $a=\frac{v-u}{t}=\frac{15-0}{5}=3{\text{m/s}}^2$.
2. $s=ut+\frac{1}{2}a{t}^2=0+\frac{1}{2}\times 3\times 5^2=37.5\text{m}$.

Visual:

Real-World Tie: Pathao’s algorithm optimizes delivery routes using kinematic principles to minimize time (and fuel) for riders.

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2. Dynamics: Forces and Newton’s Laws

Forces cause changes in motion. Newton’s three laws form the foundation of classical mechanics.

Newton’s Laws:

1. First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net force.
   • Example: A passenger lurches forward when a bus brakes suddenly (their body resists change in motion).
2. Second Law: $F_{\text{net}}=ma$ (Force = mass × acceleration).
   • Example: Pushing a Daraz delivery cart harder increases its acceleration.
3. Third Law: For every action, there’s an equal and opposite reaction.
   • Example: A rocket expels gas downward; the gas pushes the rocket upward.

Types of Forces:

Visual:

Worked Example: Khalti App’s Server Load

A Khalti server (mass = 500 kg) experiences a frictional force of 200 N while moving horizontally. If a force of 600 N is applied, calculate:

1. Net force and acceleration.
2. Velocity after 10 seconds (assuming $u=0$).

Solution:

1. $F_{\text{net}}=F_{\text{applied}}-f=600-200=400\text{N}$. $a=\frac{F_{\text{net}}}{m}=\frac{400}{500}=0.8{\text{m/s}}^2$.
2. $v=u+at=0+0.8\times 10=8\text{m/s}$.

Real-World Tie: Khalti’s servers must handle sudden load spikes (like Diwali sales) without crashing—Newton’s laws help engineers design stable systems.

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3. Hooke’s Law and Simple Harmonic Motion (SHM)

SHM is periodic motion where the restoring force is proportional to displacement.

Hooke’s Law:

$$F=-kx$$

• $k$ = spring constant (N/m), $x$ = displacement from equilibrium.
• Negative sign: Force opposes displacement (restoring force).

Visual:

Differential Equation of SHM:

From Newton’s second law: $$ma=-kx⟹\frac{d^{2}x}{d{t}^2}+\frac{k}{m}x=0$$ General Solution: $$x(t)=A\cos (\omega t+\varphi )$$ where:

• $A$ = amplitude (maximum displacement),
• $\omega =\sqrt{\frac{k}{m}}$ = angular frequency (rad/s),
• $\varphi$ = phase angle.

Key Quantities in SHM:

Worked Example: eSewa Bill Payment Spring

A spring in an eSewa payment terminal stretches 0.05 m when a 10 N force is applied. A 0.5 kg mass is attached and set oscillating.

1. Find the spring constant $k$.
2. Calculate the period of oscillation.
3. If released from $x=0.1\text{m}$, find the velocity at $x=0.05\text{m}$.

Solution:

1. $k=\frac{F}{x}=\frac{10}{0.05}=200\text{N/m}$.
2. $T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{0.5}{200}}=0.314\text{s}$.
3. Use energy conservation: $$\frac{1}{2}k{A}^2=\frac{1}{2}k{x}^2+\frac{1}{2}m{v}^2$$ At $x=0.05\text{m}$, $v=\sqrt{\frac{k}{m}(A^2-x^2)}=\sqrt{\frac{200}{0.5}(0.1^2-0.05^2)}=1.41\text{m/s}$.

Real-World Tie: eSewa’s mechanical components (like buttons) rely on SHM for reliable operation—springs ensure consistent force application.

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4. Energy in Simple Harmonic Motion

Total energy in SHM is conserved and alternates between kinetic and potential.

Energy Equations:

• Potential Energy (PE): $U=\frac{1}{2}k{x}^2$.
• Kinetic Energy (KE): $K=\frac{1}{2}m{v}^2$.
• Total Energy (E): $E=\frac{1}{2}k{A}^2$ (constant).

Graph of Energy vs. Time:

Worked Example: NTC Power Grid Oscillations

A 2 kg generator component oscillates with amplitude 0.2 m and spring constant 50 N/m.

1. Calculate total energy.
2. Find velocity when displacement is 0.1 m.

Solution:

1. $E=\frac{1}{2}k{A}^2=\frac{1}{2}\times 50\times 0.2^2=1\text{J}$.
2. $v=\sqrt{\frac{k}{m}(A^2-x^2)}=\sqrt{\frac{50}{2}(0.2^2-0.1^2)}=1\text{m/s}$.

Real-World Tie: Power grids use SHM principles to model oscillations in transformers—preventing energy loss and ensuring stability.

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5. Damped and Forced Oscillations

Real-world oscillators lose energy to friction/air resistance (damping) and can be driven by external forces.

Damped Oscillation:

$$m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$$

• $c$ = damping coefficient.
• Under-damped: Oscillates with decreasing amplitude.
• Critically damped: Returns to equilibrium fastest without oscillating.
• Over-damped: Slow return to equilibrium.

Visual:

Forced Oscillations and Resonance:

• Driving force: $F=F_0\cos ({\omega }_{d}t)$.
• Resonance: Amplitude becomes very large when ${\omega }_d\approx \omega$ (natural frequency).
  • Example: A bridge collapsing due to marching soldiers’ frequency matching its natural frequency.

Worked Example: Kathmandu Traffic Lights A traffic light’s suspension system has $m=1\text{kg}$, $k=100\text{N/m}$, and damping $c=2\text{N⋅s/m}$. If driven at ${\omega }_d=10\text{rad/s}$:

1. Is the system under-damped, critically damped, or over-damped?
2. Find the natural frequency $\omega$.

Solution:

1. Damping ratio: $\zeta =\frac{c}{2\sqrt{mk}}=\frac{2}{2\sqrt{1\times 100}}=0.1$ (under-damped).
2. $\omega =\sqrt{\frac{k}{m}}=\sqrt{100}=10\text{rad/s}$.

Real-World Tie: Traffic light poles are designed to avoid resonance with wind or vehicle vibrations to prevent failure.

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6. Waves: Transverse and Longitudinal

Waves transfer energy without transferring matter.

Types of Waves:

Wave Equation:

$$y(x,t)=A\sin (kx-\omega t+\varphi )$$

• $A$ = amplitude,
• $k$ = wave number ($k=\frac{2\pi }{\lambda }$),
• $\omega$ = angular frequency,
• $\lambda$ = wavelength,
• $v=\frac{\omega }{k}=f\lambda$ (wave speed).

Visual:

Worked Example: YouTube Video Compression

A sound wave in a YouTube video has frequency 440 Hz and wavelength 0.78 m in air. Calculate:

1. Wave speed.
2. Time for one complete oscillation (period).

Solution:

1. $v=f\lambda =440\times 0.78=343.2\text{m/s}$ (speed of sound in air).
2. $T=\frac{1}{f}=\frac{1}{440}\approx 0.0023\text{s}$.

Real-World Tie: YouTube compresses audio/video using wave physics to reduce file size while preserving quality.

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7. Superposition and Standing Waves

When two waves meet, their displacements add (superposition). Standing waves form from interference.

Standing Waves:

• Nodes: Points with zero amplitude.
• Antinodes: Points with maximum amplitude.
• Conditions for standing waves:
  • String fixed at both ends: ${\lambda }_n=\frac{2L}{n}$ (where $n=1,2,3,...$).
  • Open pipe: ${\lambda }_n=\frac{2L}{n}$.
  • Closed pipe: ${\lambda }_n=\frac{4L}{n}$.

Visual:

Worked Example: Guitar String

A guitar string (length $L=0.6\text{m}$) vibrates in its 3rd harmonic. If the speed of waves on the string is 200 m/s:

1. Find the wavelength of the 3rd harmonic.
2. Calculate the frequency.

Solution:

1. For the 3rd harmonic ($n=3$): ${\lambda }_3=\frac{2L}{3}=\frac{1.2}{3}=0.4\text{m}$.
2. $f=\frac{v}{\lambda }=\frac{200}{0.4}=500\text{Hz}$.

Real-World Tie: Guitar makers tune strings using standing wave frequencies to produce specific notes.

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In the Real World

1. eSewa’s Payment Terminals:

   • Hooke’s Law: Springs in buttons ensure consistent force application for reliable transactions.
   • SHM: Micro-vibrations in sensors detect user input accurately.

2. Pathao’s Delivery Optimization:

   • Kinematics: Algorithms use velocity/acceleration data to optimize rider routes, reducing fuel consumption and delivery time.
   • Damping: Suspension systems in delivery bikes use dampers to absorb road shocks, protecting cargo.

3. NTC’s Power Grid Stability:

   • Resonance: Engineers design transformers to avoid resonant frequencies that could cause equipment failure during load spikes.
   • Waves: Power lines transmit AC current as electromagnetic waves, with frequency (50/60 Hz) carefully controlled to match appliances.

4. Nepal Rastra Bank’s ATM Machines:

   • SHM: Springs in card slots ensure smooth insertion/removal.
   • Energy Conservation: Kinetic energy of the card is converted to potential energy in the slot’s mechanism.

5. YouTube’s Audio Compression:

   • Wave Physics: Audio files are compressed by analyzing frequency components (Fourier transforms), reducing data size while preserving sound quality.

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Exam Tip

1. Memorize Key Formulas:

   • Newton’s second law: $F=ma$.
   • Hooke’s law: $F=-kx$.
   • SHM period: $T=2\pi \sqrt{\frac{m}{k}}$.
   • Wave speed: $v=f\lambda$.
   • Energy in SHM: $E=\frac{1}{2}k{A}^2$.

2. Unit Consistency:

   • Always convert units to SI (kg, m, s) before plugging into equations. For example, if $k$ is given in N/cm, convert to N/m.

3. Graphical Questions:

   • For SHM, sketch displacement-time, velocity-time, and acceleration-time graphs. Remember:
     • Displacement leads velocity by 90°.
     • Velocity leads acceleration by 90°.
   • For waves, label amplitude, wavelength, and direction of propagation.

4. Real-World Applications:

   • Examiners often ask to relate concepts to everyday objects (e.g., "Explain how a car suspension works using SHM").
   • Example Answer: "A car’s shock absorber uses a damping mechanism to reduce oscillations caused by bumps. The spring follows Hooke’s law ($F=-kx$), while the damper introduces a velocity-dependent force ($F=-cv$) to prevent excessive bouncing, ensuring a smooth ride."

5. Differential Equations:

   • For SHM, start with $F=ma$ and substitute $F=-kx$ to derive $\frac{d^{2}x}{d{t}^2}+\frac{k}{m}x=0$. Recognize this as the standard SHM equation.
   • For damped oscillations, include the damping term $c\frac{dx}{dt}$.

6. Numerical Problems:

   • Break questions into steps:
     1. Identify given quantities and what’s asked.
     2. Write down relevant formulas.
     3. Substitute values with units.
     4. Solve step-by-step, showing all working.
   • Example: If a spring stretches 0.1 m under 20 N, calculate $k$ first ($k=\frac{F}{x}=200\text{N/m}$) before finding $T$.

7. Diagrams:

   • Always draw free-body diagrams for dynamics problems (show all forces acting on the object).
   • For SHM, label equilibrium position, amplitude, and direction of motion.

8. Common Pitfalls:

   • Sign Errors: In SHM, the restoring force is negative ($F=-kx$). Forgetting the negative sign leads to incorrect direction.
   • Units: Mixing up radians and degrees in angular frequency ($\omega$ is in rad/s).
   • Assumptions: For SHM, assume no damping unless stated. For waves, confirm if the medium is fixed or free at boundaries.

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Final Note: Mechanics and oscillations are the backbone of physics. Master these concepts, and you’ll excel in both exams and real-world problem-solving—from designing bridges to optimizing delivery routes!