2026-08-19 19:01:21: Rotational Inertia

2026-08-19 19:01:21 -04:00
parent 9f5e740d87
commit fb463d0b5f
8 changed files with 143 additions and 16 deletions
@@ -8,3 +8,15 @@
 * Impulse units are N * s (kg * m/s)
 * When mass is constant F=ma comes from Impulse-Momentum theory

+|Term|Definition|
+|---|---|
+|axis of rotation|The fixed line about which a system rotates.|
+|force component perpendicular|The portion of an applied force that acts at a right angle to the position vector from the axis of rotation.|
+|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.|
+|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.|
+|lever arm|The perpendicular distance from the axis of rotation to the line of action of an applied force.|
+|line of action|The straight line along which a force acts, extending infinitely in both directions.|
+|perpendicular force|The component of a force that is perpendicular to the position vector, which directly contributes to torque production.|
+|position vector|A vector drawn from the axis of rotation to the point where a force is applied on a rigid system.|
+|rigid system|A system that holds its shape but in which different points on the system move in different directions during rotation.|
+|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.|
@@ -8,7 +8,7 @@
 * "Lost" kinetic energy just transforms into different forms, so total energy is preserved.

 Perfectly inelastic collisions:
-$$v_f=\frac{m_1v_1+m_2v_2}{m_1m_2}$$
+$$v_f=\frac{m_1v_1+m_2v_2}{m_1+m_2}$$
 ![[Pasted image 20260819112153.png]]
 # Elastic Collision Energy Conservation Formula
 $$\frac{1}{2}m_1v^2_{1i}+\frac{1}{2}m_2v^2_{2i}=\frac{1}{2}m_1v^2_{1f}+\frac{1}{2}m_2v^2_{2f}$$
@@ -0,0 +1,6 @@
+[[Torque and Rotational Dynamics]]
+
+# Summary
+* Linear velocity related to angular velocity: $$v=rw$$ v= velocity r=how far w=angular velocity
+* Tangential acceleration relates to linear acceleration: $$A_T=ra$$
+* Arc length and angle linked by: $$s=r\theta$$
@@ -0,0 +1,9 @@
+[[Torque and Rotational Dynamics]]
+For a point mass:$$I=mr^2$$
+$I$ is rotational inertia (kg * $m^2$)
+m is mass
+r is radial distance
+
+Several discrete objects:
+$$I_{tot}=\Sigma I_i=\Sigma m_i r^2$$
+
@@ -0,0 +1,18 @@
+[[Torque and Rotational Dynamics]]
+
+# Summary
+* Angular displacement is measured with $\Delta \theta=\theta-\theta_0$
+* Track angular displacement direction with clockwise counterclockwise sign convention
+* Average angular velocity: $w_{avg}=\frac{\Delta\theta}{\Delta t}$
+* Average angular acceleration: $a_{avg}=\frac{\Delta w}{\Delta t}$
+* 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
+* Slope of theta vs time shows angular velocity
+* Slope of angular velocity vs time shows angular acceleration
+* Integral of angular velocity vs time shows Delta theta
+
+# Angular vs. Linear Motion
+Angular motion equations are similar to linear motion equations. 
+$w=w_0+at$
+$\theta=\theta_0+w_0t+\frac{1}{2}at^2$
+$w^2=w^2_0+2a(\theta - \theta_0)$
+
@@ -0,0 +1 @@
+[[physics]]
@@ -0,0 +1,10 @@
+[[Torque and Rotational Dynamics]]
+
+Twisting effect a force has around an axis of rotation.
+
+# Summary
+* Only the force component perpendicular to $\vec r$ generates torque.
+* Magnitude of torque: $$\tau=rF\sin \theta=rF_\perp$$
+* Torque is at its largest when the force is applied 90$\degree$ to r.
+* Force diagrams are like FBD but shows how each force acts relative to the axis.
+