#periodic motion

Forced Oscillations and Resonance

Which of the following periodic motions cannot be classified as oscillatory due to the absence of a fixed equilibrium po

**Forced oscillations** result when external periodic driving force F = F₀ cos(ω_d t) acts on oscillator, steady-state frequency equals driving frequency ω_d, amplitude A = F₀/√((k - m ω_d²)² + (b ω_d)²) depends on proximity to natural frequency ω₀ = √(k/m). Resonance when ω_d ≈ ω₀, amplitude maximum. Rotational motion of a ceiling fan is periodic but not oscillatory, as it lacks a fixed equilibrium point about which it moves to-and-fro, unlike SHM examples. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result Rotational motion of a ceiling fan follows, reflecting

Ref: NCERT > Physics Book > Oscillations > Forced Oscillations and Resonance

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Forced Oscillations and Resonance

What underlies the periodic nature of SHM when expressed as a superposition of sine and cosine functions?

**Resonance phenomenon** amplifies response when driving frequency matches natural frequency ω_d ≈ ω₀, large amplitude even with small F₀, as damping limits growth. Natural frequency determined by system parameters, resonance condition crucial for understanding vibrations and energy absorption, e.g., bridge collapse, tuning. The periodicity arises from the repeating nature of sine and cosine functions, which have a fixed period ( 2π/ω ), ensuring the motion repeats consistently. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The periodicity of trigonometric functions follows, reflecting SHM dependence on amplitude A, ω and system parameters.

Ref: NCERT > Physics Book > Oscillations > Forced Oscillations and Resonance

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Damped Oscillations

Which of the following represents periodic motion but not SHM? (\( \omega \) is a positive constant)

**Damped oscillations** occur when resistive forces dissipate energy, amplitude decays exponentially as A(t) = A₀ e^(-b t/2m), b damping coefficient (kg/s), frequency slightly reduced ω' = √(ω₀² - (b/2m)²). Damping arises from friction or viscosity, energy loss per cycle proportional to velocity squared, motion eventually stops. (a) 2 cos (ω t) : SHM. (b) cos ω t + cos 3ω t : Periodic (period (2π/ω) ), not SHM (multiple frequencies). (c) 3 sin (2ω t) : SHM. (d) e⁻ω t : Not periodic. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A²,

Ref: NCERT > Physics Book > Oscillations > Damped Oscillations

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Damped Oscillations

What distinguishes the oscillatory nature of a pendulum from the periodic motion of a planet in orbit?

**Damped oscillations** occur when resistive forces dissipate energy, amplitude decays exponentially as A(t) = A₀ e^(-b t/2m), b damping coefficient (kg/s), frequency slightly reduced ω' = √(ω₀² - (b/2m)²). Damping arises from friction or viscosity, energy loss per cycle proportional to velocity squared, motion eventually stops. A pendulum oscillates about a fixed point due to a linear restoring force (gravity component), while a planet’s orbit is periodic but governed by inverse-square gravitational force. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The restoring force is linear follows, reflecting SHM dependence

Ref: NCERT > Physics Book > Oscillations > Damped Oscillations

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Equations of SHM, Phase and Angular Frequency

Which statement best explains why uniform circular motion is not considered oscillatory despite being periodic?

**Angular frequency** ω = 2πf = 2π/T characterizes rapidity, independent of amplitude for SHM. Phase constant φ shifts sine/cosine, allowing any initial condition, e.g., x(0)=0 requires φ=0 for sine form. Displacement, velocity, acceleration share ω but differ in phase by 90° and 180°. Oscillatory motion requires to-and-fro movement about an equilibrium, while uniform circular motion involves continuous rotation without reversing direction. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result It does not involve to-and-fro motion follows, reflecting SHM dependence on amplitude A, ω and system parameters.

Ref: NCERT > Physics Book > Oscillations > Equations of SHM, Phase and Angular Frequency

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Equations of SHM, Phase and Angular Frequency

Which of the following is a non-periodic motion? (\( \omega \) is a positive constant)

**Angular frequency** ω = 2πf = 2π/T characterizes rapidity, independent of amplitude for SHM. Phase constant φ shifts sine/cosine, allowing any initial condition, e.g., x(0)=0 requires φ=0 for sine form. Displacement, velocity, acceleration share ω but differ in phase by 90° and 180°. (a) sin ω t + cos ω t : Periodic, period (2π/ω) . (b) sin³ ω t : Periodic, period (2π/3ω) . (c) e⁻ω t : Non-periodic (decays to zero). (d) cos 2ω t : Periodic, period (π/ω) . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result

Ref: NCERT > Physics Book > Oscillations > Equations of SHM, Phase and Angular Frequency

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Energy in SHM - Kinetic, Potential and Total

Which periodic motion lacks a restoring force directed towards a fixed equilibrium point?

**Energy distribution** shows maximum kinetic at equilibrium and maximum potential at extremes, sum constant. This relation enables calculation of amplitude, velocity at any displacement via v = ±√(2(E-U)/m), and understanding of energy storage in oscillating system for NEET problems. The rotation of a carousel is periodic but not oscillatory, as it involves continuous circular motion without a restoring force towards a fixed point, unlike SHM systems. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The rotation of a carousel follows, reflecting SHM dependence on amplitude A, ω and system parameters.

Ref: NCERT > Physics Book > Oscillations > Energy in SHM - Kinetic, Potential and Total

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Periodic Motion and SHM Basic Concepts

What is a distinguishing feature of oscillatory motion compared to general periodic motion?

**Distinction between periodic and oscillatory** clarifies all SHM is periodic but not all periodic is SHM. SHM requires linear restoring force and inertia, a ∝ -x, with ω = √(k/m). Functions like sin²ωt have period π/ω but lack a = -ω² x, thus periodic not SHM, while uniform circular motion is periodic without linear oscillation. Oscillatory motion involves to-and-fro movement about a mean position (e.g., SHM), while periodic motion (e.g., circular) repeats but may not oscillate about a fixed point. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result It involves

Ref: NCERT > Physics Book > Oscillations > Periodic Motion and SHM Basic Concepts

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Periodic Motion and SHM Basic Concepts

Which of the following represents periodic motion but not SHM? (\( \omega \) is a positive constant)

**SHM condition** is linear restoring force and inertia producing sinusoidal time dependence. Motions violating a = -ω² x, such as uniform circular motion, are periodic without oscillation about fixed point, highlighting classification criteria for NCERT. (a) 4 cos (ω t) : SHM. (b) sin ω t + sin 2ω t : Periodic (period (2π/ω) ), not SHM (multiple frequencies). (c) 3 sin (ω t - (π/3)) : SHM. (d) eω t : Not periodic. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result sin ω t + sin 2ω t follows,

Ref: NCERT > Physics Book > Oscillations > Periodic Motion and SHM Basic Concepts

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Periodic Motion and SHM Basic Concepts

Which of the following motions is periodic but does not exhibit oscillatory behavior about a fixed point?

**Distinction between periodic and oscillatory** clarifies all SHM is periodic but not all periodic is SHM. SHM requires linear restoring force and inertia, a ∝ -x, with ω = √(k/m). Functions like sin²ωt have period π/ω but lack a = -ω² x, thus periodic not SHM, while uniform circular motion is periodic without linear oscillation. Uniform circular motion is periodic (repeats after a fixed time) but not oscillatory, as it does not involve to-and-fro motion about a mean position, unlike SHM examples. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result

Ref: NCERT > Physics Book > Oscillations > Periodic Motion and SHM Basic Concepts

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