2026-08-20 14:32:36: SHM

physics/Oscillations/Defining Simple Harmonic Motion (SHM).md

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+[[Oscillations]]
+It's when a restoring force is proportional to the displacement from equilibrium.
+
+# Summary
+* $F_x=-k\Delta x$
+* Acceleration always points opposite to displacement.
+* Equilibrium is where net force is zero.
+* Maximum displacement is amplitude.
+* Velocity is maximum at equilibrium.
+* Pendulum acts as SHM at low angles because restoring force is proportional to angle.
+

physics/Oscillations/Energy of Simple Harmonic Oscillators.md

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+[[Oscillations]]
+In an ideal SHM, total mechanical energy stays constant. Kinetic energy peaks at equilibrium, potential peaks at turning points.
+
+# Summary
+* Total mechanical energy in an SHM = $E_{total}=U+K$
+* Spring-object system: $E_{total}=\frac{1}{2}kA^2$
+* Real systems lose energy to friction and such, lowering oscillation over time.
+
+
+

physics/Oscillations/Frequency and Period of SHM.md

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+[[Oscillations]]
+
+
+# Summary
+* Period is how long it takes in seconds, frequency is how many times in a second.
+* $T=1/f$
+* Mass spring oscillator has period $T=2\pi\sqrt{m/k}$
+* Small angle pendulum has period $T=2\pi\sqrt{l/g}$
+* Pendulum formula only works on small angles (~15$\degree$), larger angles increase period.
+

physics/Oscillations/Oscillations.md

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+[[physics]]

physics/Oscillations/Representing and Analyzing SHM.md

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+[[Oscillations]]
+# Summary
+* $x=A\cos(2\pi ft)$ for when object starts at max displacement (cos(0) = 1)
+* $x=A\sin(2\pi ft)$ for when object starts at equilibrium (sin(0) = 0)
+* Velocity is largest at equilibrium, acceleration is largest at turning points.
+* Amplitude does not affect period.
+* Velocity time graphs are shifted by a quarter period from displacement.
+* Acceleration time graphs are shifted by half a period.
+* Reading zeros and extrema tells position, direction of motion, and whether the object is speeding up or slowing down.
+

physics/Torque and Rotational Dynamics/Newton's Second Law in Rotational Form.md

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+[[Torque and Rotational Dynamics]]
+
+# Summary
+* Core equation: $a_{sys}=\frac{\tau_{net}}{I_{sys}}$ 
+* Derived from F=ma
+* Angular acceleration is inversely proportional to rotational inertia.
+* Angular acceleration is directly proportional to torque.

physics/Torque and Rotational Dynamics/Rotational Equilibrium and Newton's First Law in Rotational Form.md

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+[[Torque and Rotational Dynamics]]
+# Summary
+* Rotational equilibrium is keeping angular velocity constant because net torque is zero.
+* Rotational equilibrium and translational equilibrium aren't related.
+* Use force and FBDs to find all torques.
+

physics/Torque and Rotational Dynamics/Rotational Inertia.md

@@ -7,3 +7,9 @@ r is radial distance
 Several discrete objects:
 $$I_{tot}=\Sigma I_i=\Sigma m_i r^2$$
 
+# Parallel Axis Theorem
+$$I'=I_{cm}+Md^2$$
+- $I$' is rotational inertia about the parallel axis (kg⋅m²)
+- $I_{cm}$ is rotational inertia about the center-of-mass axis (kg⋅m²)
+- $M$ is the total mass of the system (kg)
+- $d$ is the perpendicular distance between the two parallel axes (m)