Physics
12
Simple Harmonic Motion
11
⚡ Quick Summary
This section discusses the energy of a block-spring system undergoing SHM and introduces angular simple harmonic motion, deriving equations for angular displacement, velocity, time period, and frequency.
['α = Γ/I = - (k/I)θ', 'd²θ/dt² = -ω²θ', 'ω = √(k/I)', 'θ = θ₀ sin(ωt + δ)', 'Ω = dθ/dt = θ₀ω cos(ωt + δ)', 'T = 2π/ω = 2π√(I/k)', 'ν = 1/T = (1/2π)√(k/I)', 'ω = √(k/I)']
- Energy in SHM (Block-Spring System):
- At mean position (x=0), potential energy is zero and kinetic energy is maximum (1/2 * m * omega^2 * A^2).
- At extreme positions (x = ±A), kinetic energy is zero and potential energy is maximum (1/2 * k * A^2 = 1/2 * m * omega^2 * A^2).
- Angular Simple Harmonic Motion:
- Conditions for a body to execute angular SHM:
- A position where the resultant torque is zero (mean position, θ=0).
- A resultant torque proportional to the angular displacement (Γ = -kθ).
- The torque acts to restore the body towards the mean position.
Torsional Oscillations
11
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A disc fixed to a wire oscillates when twisted and released. The time period depends on the disc's moment of inertia and the wire's torsional constant.
['T = 2π√(I/k)', 'k = (4π²I) / T²']
When a uniform disc is fixed at its center to a metal wire and rotated about the wire, it undergoes torsional oscillations. The time period of these oscillations is given by:
`T = 2π√(I/k)`
where:
* `T` is the time period of oscillation
* `I` is the moment of inertia of the disc about the wire
* `k` is the torsional constant of the wire.
Composition of Two Simple Harmonic Motions
11
⚡ Quick Summary
When a particle experiences two forces that would each cause simple harmonic motion, the resulting motion is a combination of both. The position of the particle is the sum of the positions it would have if each force acted alone.
['r = r₁ + r₂', 'u = u₁ + u₂']
If a particle is acted upon by two separate forces, each capable of producing simple harmonic motion, the resultant motion is a combination of the two. Let `F₁` and `F₂` be the two forces. If `r₁` is the position of the particle under the influence of `F₁` alone, and `r₂` is the position under `F₂` alone, then the resultant position `r` is given by:
`r = r₁ + r₂`
Similarly, the resultant velocity `u` is given by:
`u = u₁ + u₂`
This holds true if these conditions are met at `t = 0`.
Composition of Two Simple Harmonic Motions in the Same Direction
11
⚡ Quick Summary
When two SHM's with the same frequency but possibly different amplitudes and phases act on a particle along the same direction, the resulting motion is also SHM. The resultant displacement is the sum of individual displacements.
['x = x₁ + x₂', 'x₁ = A₁sin(ωt)', 'x₂ = A₂sin(ωt + δ)', 'x = (A₁ + A₂cosδ)sin(ωt) + (A₂sinδ)cos(ωt)']
Suppose two forces act on a particle, each capable of producing simple harmonic motion along the x-direction:
`x₁ = A₁sin(ωt)`
`x₂ = A₂sin(ωt + δ)`
where `A₁` and `A₂` are the amplitudes, `ω` is the angular frequency, and `δ` is the phase difference. The resultant position `x` is given by:
`x = x₁ + x₂ = A₁sin(ωt) + A₂sin(ωt + δ)`
Expanding and simplifying:
`x = (A₁ + A₂cosδ)sin(ωt) + (A₂sinδ)cos(ωt)`