2026-08-19 19:01:21: Rotational Inertia
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@@ -8,3 +8,15 @@
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* Impulse units are N * s (kg * m/s)
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* When mass is constant F=ma comes from Impulse-Momentum theory
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|Term|Definition|
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|axis of rotation|The fixed line about which a system rotates.|
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|force component perpendicular|The portion of an applied force that acts at a right angle to the position vector from the axis of rotation.|
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|force diagram|A diagram used to represent and analyze the forces and torques exerted on a rigid system, showing the magnitude, direction, and point of application of each force relative to the axis of rotation.|
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|free-body diagram|A visual representation that shows all forces exerted on an object or system, with each force drawn as a vector originating from the object's center of mass.|
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|lever arm|The perpendicular distance from the axis of rotation to the line of action of an applied force.|
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|line of action|The straight line along which a force acts, extending infinitely in both directions.|
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|perpendicular force|The component of a force that is perpendicular to the position vector, which directly contributes to torque production.|
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|position vector|A vector drawn from the axis of rotation to the point where a force is applied on a rigid system.|
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|rigid system|A system that holds its shape but in which different points on the system move in different directions during rotation.|
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|torque|A measure of the rotational effect of a force on a rigid system, calculated as the product of the force and its perpendicular distance from the axis of rotation.|
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@@ -8,7 +8,7 @@
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* "Lost" kinetic energy just transforms into different forms, so total energy is preserved.
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Perfectly inelastic collisions:
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$$v_f=\frac{m_1v_1+m_2v_2}{m_1m_2}$$
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$$v_f=\frac{m_1v_1+m_2v_2}{m_1+m_2}$$
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![[Pasted image 20260819112153.png]]
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# Elastic Collision Energy Conservation Formula
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$$\frac{1}{2}m_1v^2_{1i}+\frac{1}{2}m_2v^2_{2i}=\frac{1}{2}m_1v^2_{1f}+\frac{1}{2}m_2v^2_{2f}$$
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@@ -0,0 +1,6 @@
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[[Torque and Rotational Dynamics]]
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# Summary
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* Linear velocity related to angular velocity: $$v=rw$$ v= velocity r=how far w=angular velocity
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* Tangential acceleration relates to linear acceleration: $$A_T=ra$$
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* Arc length and angle linked by: $$s=r\theta$$
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@@ -0,0 +1,9 @@
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[[Torque and Rotational Dynamics]]
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For a point mass:$$I=mr^2$$
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$I$ is rotational inertia (kg * $m^2$)
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m is mass
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r is radial distance
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Several discrete objects:
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$$I_{tot}=\Sigma I_i=\Sigma m_i r^2$$
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[[Torque and Rotational Dynamics]]
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# Summary
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* Angular displacement is measured with $\Delta \theta=\theta-\theta_0$
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* Track angular displacement direction with clockwise counterclockwise sign convention
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* Average angular velocity: $w_{avg}=\frac{\Delta\theta}{\Delta t}$
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* Average angular acceleration: $a_{avg}=\frac{\Delta w}{\Delta t}$
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* Rigid system holds shape, but different parts move at different speeds so you can't treat it as a single particle unless COM motion describes rotation well
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* Slope of theta vs time shows angular velocity
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* Slope of angular velocity vs time shows angular acceleration
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* Integral of angular velocity vs time shows Delta theta
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# Angular vs. Linear Motion
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Angular motion equations are similar to linear motion equations.
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$w=w_0+at$
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$\theta=\theta_0+w_0t+\frac{1}{2}at^2$
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$w^2=w^2_0+2a(\theta - \theta_0)$
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@@ -0,0 +1 @@
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[[physics]]
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@@ -0,0 +1,10 @@
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[[Torque and Rotational Dynamics]]
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Twisting effect a force has around an axis of rotation.
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# Summary
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* Only the force component perpendicular to $\vec r$ generates torque.
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* Magnitude of torque: $$\tau=rF\sin \theta=rF_\perp$$
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* Torque is at its largest when the force is applied 90$\degree$ to r.
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* Force diagrams are like FBD but shows how each force acts relative to the axis.
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