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झारखण्ड बोर्ड (JAC) • कक्षा XI • Physics • अध्याय 13

अनुमानित समय: 45 Mins

प्रगति: अध्ययनरत

दोलन (Oscillations) (Oscillations)

In Class 11 Physics, "Oscillations" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें

प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

🕰️ Have You Ever Wondered?

Why does a grandfather clock keep perfect rhythmic time for centuries using a simple swinging pendulum, or why do marching soldiers break cadence when crossing bridges to avoid collapsing them? Simple Harmonic Motion (SHM) and Resonance govern all vibrating systems.

यह अध्याय क्यों महत्वपूर्ण है

In Class 11 Physics, "Oscillations" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्ययन से पूर्व (आवश्यक ज्ञान)

Kinematics and forces.
Sine and cosine functions.
Conservation of mechanical energy.

इस अध्याय के लक्ष्य

Define Periodic motion, Oscillatory motion, and Simple Harmonic Motion (SHM: $a = -\omega^2 x$).
Write displacement equation of SHM: $x(t) = A\cos(\omega t + \phi)$ and find velocity ($v = \pm\omega\sqrt{A^2 - x^2}$) and acceleration.
Analyze Energy in SHM: Kinetic energy ($K = \frac{1}{2}m\omega^2(A^2 - x^2)$), Potential energy ($U = \frac{1}{2}m\omega^2 x^2$), and Total Energy ($E = \frac{1}{2}m\omega^2 A^2 = \text{constant}$).
Derive time period of a Simple Pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$ and a Spring-Mass system: $T = 2\pi\sqrt{\frac{m}{k}}$.
Distinguish between Free, Damped, and Forced Oscillations; explain the catastrophic phenomenon of Resonance.

अध्याय रूपरेखा एवं प्रगति

1 1. The Mathematics of Simple Harmon...

2 2. Energy Conservation in SHM

3 3. The Simple Pendulum & Resonance

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. The Mathematics of Simple Harmonic Motion

SHM is oscillatory motion where acceleration is directly proportional to displacement and directed towards the mean position: $$\mathbf{F = -k x \implies a = -\omega^2 x} \quad \left(\omega = \sqrt{\frac{k}{m}}\right)$$
• Displacement: $\mathbf{x(t) = A\cos(\omega t + \phi)}$.
• Velocity: $v(t) = \frac{dx}{dt} = -A\omega\sin(\omega t + \phi) = \mathbf{\pm \omega\sqrt{A^2 - x^2}}$ (Max $v_{\text{max}} = A\omega$ at center!).
• Acceleration: $a(t) = -A\omega^2\cos(\omega t + \phi) = \mathbf{-\omega^2 x}$ (Max $a_{\text{max}} = \omega^2 A$ at extreme ends!).

2. Energy Conservation in SHM

Energy oscillates seamlessly between kinetic and potential forms:
• Kinetic Energy: $K = \frac{1}{2}m\omega^2(A^2 - x^2)$ (maximum at $x = 0$).
• Potential Energy: $U = \frac{1}{2}m\omega^2 x^2$ (maximum at extremes $x = \pm A$).
• Total Energy: $$\mathbf{E = K + U = \frac{1}{2}m\omega^2 A^2 = \text{constant}} \quad (E \propto A^2)$$

3. The Simple Pendulum & Resonance

Simple Pendulum: Restoring torque $\tau = -mg L \sin\theta \approx -mg L \theta$. Time period: $$\mathbf{T = 2\pi\sqrt{\frac{L}{g}}} \quad (\text{Independent of mass and amplitude!})$$
Resonance: When driving frequency matches the natural natural frequency of a system ($\omega_d = \omega_0$), energy transfer is maximized, producing gigantic catastrophic vibration amplitudes!

Oscillations - Key Conceptual & Analytical Model

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

SHM Condition: Linear restoring force $F = -kx$ yielding acceleration $a = -\omega^2 x$.

मुख्य बिंदु 2

Phase Quadrature: Velocity leads displacement by $90^\circ$; acceleration is $180^\circ$ opposite.

मुख्य बिंदु 3

Energy Invariance: Total energy $E = \frac{1}{2}m\omega^2 A^2$ conserved through quadratic handoffs.

मुख्य बिंदु 4

Pendulum Formula: $T = 2\pi\sqrt{L/g}$ independent of bob mass or displacement angle.

मुख्य बिंदु 5

Resonance: Maximum amplitude vibration when driving frequency equals natural frequency.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

A particle executes SHM with an amplitude of 4 cm. At what displacement from the mean position is its kinetic energy equal to its potential energy?

उत्तर एवं व्याख्या देखें

उत्तर: $K = U \implies \frac{1}{2}m\omega^2(A^2 - x^2) = \frac{1}{2}m\omega^2 x^2 \implies A^2 - x^2 = x^2 \implies 2x^2 = A^2 \implies x = \pm \frac{A}{\sqrt{2}} = \pm \frac{4}{\sqrt{2}} = \pm 2\sqrt{2}\text{ cm} \approx \pm 2.83\text{ cm}$.
x = ±2√2 cm (≈ ±2.83 cm).

What is the length of a seconds pendulum (a pendulum having a period of exactly 2 seconds) on the surface of the Earth ($g = 9.8\text{ m/s}^2$)?

उत्तर एवं व्याख्या देखें

उत्तर: $T = 2\pi\sqrt{\frac{L}{g}} \implies 2 = 2\pi\sqrt{\frac{L}{9.8}} \implies 1 = \pi\sqrt{\frac{L}{9.8}} \implies 1 = \pi^2 \frac{L}{9.8} \implies L = \frac{9.8}{\pi^2} \approx \frac{9.8}{9.87} \approx 0.993\text{ m} \approx 1\text{ meter}$.
Length ≈ 1 meter (0.993 m).

Why are marching soldiers ordered to break step when crossing a suspension bridge?

उत्तर एवं व्याख्या देखें

उत्तर: If soldiers march in rhythmic lockstep, the periodic driving frequency of their footsteps may match the natural frequency of the bridge. This would trigger Resonance, building massive oscillations that can snap steel cables and collapse the bridge.
Prevents resonance that could collapse the bridge.

Write the relation between phase of displacement, velocity, and acceleration in SHM.

उत्तर एवं व्याख्या देखें

उत्तर: Velocity leads displacement in phase by $\pi/2$ ($90^\circ$); Acceleration leads velocity by $\pi/2$ and is opposite in phase to displacement by $\pi$ radians ($180^\circ$).
Velocity leads by 90°; acceleration is 180° out of phase.

How does the time period of a simple pendulum change if it is taken to the Moon?

उत्तर एवं व्याख्या देखें

उत्तर: Since $T = 2\pi\sqrt{L/g}$ and lunar gravity is $g_{\text{moon}} = g/6$, the time period increases by a factor of $\sqrt{6} \approx 2.45$; the pendulum swings much slower.
Increases by √6 times (swings slower).

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