102 lines
3.7 KiB
Markdown
102 lines
3.7 KiB
Markdown
> [!NOTE] A mass $m$ attached to a spring with constant $k$ oscillates on a frictionless surface. Derive an expression for the velocity $v$ as a function of displacement $x$.
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>
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> [!INFO]-
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> Total mechanical energy in SHM is conserved:
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> $$
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> E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2
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> $$
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> Solving for $v$:
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> $$
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> v = \pm \sqrt{\frac{k}{m}(A^2 - x^2)}
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> $$
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> [!NOTE] A wave traveling along a rope is represented by $y(x,t) = 0.02\cos(40x - 600t)$. Determine the amplitude, wavelength, frequency, and speed of the wave.
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>
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> [!INFO]-
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> The general wave form is $y = A\cos(kx - \omega t)$:
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> - Amplitude $A = 0.02\ \mathrm{m}$
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> - Wave number $k = 40 \Rightarrow \lambda = \frac{2\pi}{k} = \frac{2\pi}{40} = 0.157\ \mathrm{m}$
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> - Angular frequency $\omega = 600 \Rightarrow f = \frac{\omega}{2\pi} = \frac{600}{2\pi} \approx 95.5\ \mathrm{Hz}$
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> - Wave speed $v = f\lambda = 95.5 \times 0.157 \approx 15\ \mathrm{m/s}$
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> [!NOTE] In Young's double slit experiment, fringes are formed on a screen 1.2 m away using light of wavelength $600\ \text{nm}$. The slits are separated by $0.2\ \text{mm}$. Calculate the distance between adjacent bright fringes.
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>
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> [!INFO]-
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> Fringe spacing is given by:
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> $$
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> y = \frac{\lambda D}{d} = \frac{600 \times 10^{-9} \times 1.2}{0.2 \times 10^{-3}} = 3.6\ \text{mm}
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> $$
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> [!NOTE] A pendulum of length $0.5\ \mathrm{m}$ is displaced by a small angle. Determine its period and explain why amplitude does not affect the result.
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>
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> [!INFO]-
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> The period is:
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> $$
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> T = 2\pi\sqrt{\frac{L}{g}} = 2\pi\sqrt{\frac{0.5}{9.8}} \approx 1.41\ \mathrm{s}
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> $$
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> In the small-angle approximation ($\theta < 10^\circ$), motion is independent of amplitude.
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> [!NOTE] A charged particle $q$ moves through a uniform electric field $E$. Derive the expression for the work done on the charge and its change in potential energy.
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>
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> [!INFO]-
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> Work done:
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> $$
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> W = qEd
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> $$
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> Change in potential energy:
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> $$
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> \Delta U = -qEd
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> $$
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> since electric potential energy decreases when the charge moves in the direction of the field.
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> [!NOTE] A capacitor of $10\ \mu\mathrm{F}$ is charged to $5\ \mathrm{V}$. Calculate the energy stored in it.
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>
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> [!INFO]-
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> $$
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> E = \frac{1}{2}CV^2 = \frac{1}{2} \cdot 10 \times 10^{-6} \cdot 25 = 1.25 \times 10^{-4}\ \mathrm{J}
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> $$
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> [!NOTE] A coil of wire rotates in a magnetic field. Use Faraday’s law to derive the expression for the induced emf.
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>
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> [!INFO]-
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> Faraday’s law:
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> $$
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> \mathcal{E} = -\frac{d\Phi}{dt}
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> $$
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> Magnetic flux $\Phi = B A \cos(\omega t)$, so:
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> $$
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> \mathcal{E} = B A \omega \sin(\omega t)
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> $$
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> [!NOTE] A wire carries a current of $3\ \mathrm{A}$ through a magnetic field of $0.5\ \mathrm{T}$ perpendicular to its length. The wire is $0.4\ \mathrm{m}$ long. Find the magnetic force.
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>
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> [!INFO]-
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> $$
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> F = ILB\sin\theta = 3 \cdot 0.4 \cdot 0.5 \cdot 1 = 0.6\ \mathrm{N}
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> $$
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> [!NOTE] Explain the effect of damping on the amplitude-frequency graph of a driven harmonic oscillator.
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>
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> [!INFO]-
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> Damping reduces the peak amplitude and shifts the resonant frequency slightly lower. Greater damping broadens the curve and lowers the quality factor $Q$.
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> [!NOTE] A mass-spring oscillator experiences light damping. Write the differential equation and general solution.
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>
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> [!INFO]-
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> Equation:
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> $$
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> m\ddot{x} + b\dot{x} + kx = 0
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> $$
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> Solution:
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> $$
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> x(t) = A e^{-\gamma t} \cos(\omega' t + \phi)
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> $$
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> where $\gamma = \frac{b}{2m}$ and $\omega' = \sqrt{\omega_0^2 - \gamma^2}$.
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> [!NOTE] A $0.2\ \mathrm{kg}$ object experiences a force due to gravity from Earth at a distance of $6.4 \times 10^6\ \mathrm{m}$. Calculate the force.
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>
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> [!INFO]-
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> $$
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> F = G\frac{Mm}{r^2} = 6.67 \times 10^{-11} \cdot \frac{5.97 \times 10^{24} \cdot 0.2}{(6.4 \times 10^6)^2} \approx 1.96\ \mathrm{N}
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> $$
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