#velocity function

Equations of SHM, Phase and Angular Frequency

A mass oscillates with \( v = -12 \sin (6t) \) (in m/s). What is its amplitude?

**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°. Velocity: v = -ω A sin (ω t) . ω = 6 s⁻¹, vₘₐₓ = ω A = 12 ⇒ A = (12/6) = 2 m . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 2.0 m 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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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

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 mass oscillates with \( v = -12 \cos (4t) \) (in m/s). What is its amplitude?

**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) , but given v = -12 cos (4t) . ω = 4 s⁻¹, vₘₐₓ = ω A = 12 ⇒ A = (12/4) = 3 m . Applying x = A cos(ωt + φ), v = -ωA sin(ωt + φ), a = -ω²x and E = ½kA² = ½mω²A², result 3.0 m

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

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Displacement, Velocity and Acceleration in SHM

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

**SHM kinematics** given by x = A cos(ωt + φ), velocity v = dx/dt = -ω A sin(ωt + φ), acceleration a = dv/dt = -ω² A cos(ωt + φ) = -ω² x, maxima v_max = ωA at mean position x=0, a_max = ω²A at extremes x=±A. Phase φ determines initial position, ω = 2π/T = √(k/m). Velocity: v = -ω A sin (ω t) . 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 = -ω²x and E = ½kA² =

Ref: NCERT > Physics Book > Oscillations > Displacement, Velocity and Acceleration in SHM

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