Periodic Motion and SHM Basic Concepts

Periodic Motion and SHM Basic Concepts

What causes the restoring force in a spring-mass system to initiate SHM?

**Periodic motion** repeats after fixed period T, x(t+T)=x(t), while oscillatory motion involves to-and-fro about equilibrium. SHM is special periodic motion where restoring force proportional to displacement, F = -k x, acceleration a = -ω² x, leading to sinusoidal displacement x = A cos(ωt + φ). The elasticity of the spring generates a restoring force ( F = -kx ) proportional to displacement, driving the oscillatory motion characteristic of SHM. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result Elasticity of the spring follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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

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Periodic Motion and SHM Basic Concepts

A particle’s displacement is \( x = 3 \cos (4\pi t + \frac{\pi}{3}) \) (in m). What is its velocity at \( t = 0 \, \text

**Periodic motion** repeats after fixed period T, x(t+T)=x(t), while oscillatory motion involves to-and-fro about equilibrium. SHM is special periodic motion where restoring force proportional to displacement, F = -k x, acceleration a = -ω² x, leading to sinusoidal displacement x = A cos(ωt + φ). Velocity: v = -ω A sin (ω t + Φ) . A = 3 m, ω = 4π s⁻¹, Φ = (π/3) . At t = 0 : v = -4π × 3 sin (π/3) = -12π × (√(3)/2) ≈ -32.58 m/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and

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

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Periodic Motion and SHM Basic Concepts

In an oscillatory system, if the displacement and acceleration are always in opposite directions, what type of motion is

**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. In SHM, acceleration ( a = -ω² x ) is always opposite to displacement, a hallmark of harmonic motion due to the restoring force. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result Simple harmonic motion

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

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Periodic Motion and SHM Basic Concepts

A particle’s x-projection from circular motion is \( x = 7 \cos (3t) \) (in m). What is its maximum acceleration?

**Periodic motion** repeats after fixed period T, x(t+T)=x(t), while oscillatory motion involves to-and-fro about equilibrium. SHM is special periodic motion where restoring force proportional to displacement, F = -k x, acceleration a = -ω² x, leading to sinusoidal displacement x = A cos(ωt + φ). Maximum acceleration: aₘₐₓ = ω² A . A = 7 m, ω = 3 s⁻¹ . aₘₐₓ = 3² × 7 = 9 × 7 = 63 m/s² . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 63 m/s² follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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

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Periodic Motion and SHM Basic Concepts

A mass oscillates with \( v = -6 \sin (4t) \) (in m/s). What is its amplitude?

**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. Velocity: v = -ω A sin (ω t) . Given ω = 4 s⁻¹, vₘₐₓ = ω A = 6 . A = (6/4) = 1.5 m . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A²,

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

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Periodic Motion and SHM Basic Concepts

Which aspect of SHM ensures that the time period is independent of the initial conditions?

**Periodic motion** repeats after fixed period T, x(t+T)=x(t), while oscillatory motion involves to-and-fro about equilibrium. SHM is special periodic motion where restoring force proportional to displacement, F = -k x, acceleration a = -ω² x, leading to sinusoidal displacement x = A cos(ωt + φ). The period T = (2π/ω) depends only on ω = √((k/m)) , a constant derived from system properties, not initial displacement or velocity. Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The constant angular frequency follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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

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Periodic Motion and SHM Basic Concepts

Which characteristic of SHM ensures that the motion repeats exactly after a fixed interval?

**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. The sinusoidal nature of displacement ( x = A cos (ω t + Φ) ) ensures periodicity, as the cosine function repeats every 2π , giving a fixed period T = (2π/ω) . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result The sinusoidal variation follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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

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Periodic Motion and SHM Basic Concepts

A mass oscillates with \( v = -10 \cos (5t) \) (in m/s). What is its amplitude?

**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. Velocity: v = -ω A sin (ω t) , but given v = -10 cos (5t) . ω = 5 s⁻¹, vₘₐₓ = ω A = 10 ⇒ A = (10/5) = 2 m . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a =

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

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Periodic Motion and SHM Basic Concepts

A particle in SHM has \( x = 6 \cos (4t) \) (in m). What is its maximum acceleration?

**Periodic motion** repeats after fixed period T, x(t+T)=x(t), while oscillatory motion involves to-and-fro about equilibrium. SHM is special periodic motion where restoring force proportional to displacement, F = -k x, acceleration a = -ω² x, leading to sinusoidal displacement x = A cos(ωt + φ). Maximum acceleration: aₘₐₓ = ω² A . A = 6 m, ω = 4 s⁻¹ . aₘₐₓ = 4² × 6 = 16 × 6 = 96 m/s² . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 96 m/s² follows, reflecting SHM dependence on amplitude A, ω and system parameters.

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

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Periodic Motion and SHM Basic Concepts

A particle in SHM has \( x = 4 \sin (3t) \) (in m). What is its speed at \( x = 2 \, \text{m} \)?

**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. Velocity: v = ± ω √(A² - x²) . A = 4 m, ω = 3 s⁻¹, x = 2 m . v = 3 √(4² - 2²) = 3 √(16 - 4) = 3 √(12) = 3 × 2√(3) ≈ 10.39 m/s . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 10.39 m/s follows,

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

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Periodic Motion and SHM Basic Concepts

A particle in SHM has a displacement \( x = 3 \cos (4t + \frac{\pi}{3}) \) (in meters). What is its acceleration at \( t

**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. Acceleration: a(t) = -ω² x(t) . Here, ω = 4 s⁻¹, x(0) = 3 cos ((π/3)) = 3 × 0.5 = 1.5 m . a(0) = -4² × 1.5 = -16 × 1.5 = -24 m/s² . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ),

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

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Periodic Motion and SHM Basic Concepts

A particle in SHM has \( a = -81 x \) (in SI units). What is its frequency?

**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. For SHM, a = -ω² x . Given a = -81 x , ω² = 81 ⇒ ω = 9 rad/s . Frequency: v = (ω/2π) = (9/2 × 3.14) ≈ 1.43 Hz . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x

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

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