Angular Motion

With Center of Mass of a system of particles is when

    The total mass is concentrated at that point, and

    the net force on the system is acting through that point.

Center of Mass: Def/Formula

$\overrightarrow{r}_{\operatorname{cm}}=\frac{\sum_{i}m_{i}\left(\overrightarrow{r}_{i}\right)}{\sum_{i}m_{i}}$ | For overall center, with m standing for mass, and r standing for position.

$x_{\operatorname{cm}}=\frac{\sum_{i}m_{i}x_{i}}{\sum_{i}m_{i}}$ | For X direction

$y_{\operatorname{cm}}=\frac{\sum_{i}m_{i}y_{i}}{\sum_{i}m_{i}}$ | For Y direction

z_{\operatorname{cm}}=\frac{\sum_{i}m_{i}z_{i}}{\sum_{i}m_{i}}$| For Z direction

Basically, like the lab, to solve for this, you would consecutively add the (mass times the position of an object i, like object 1, object 2, up to i times)

Divided by the sum of all of the masses.

When it comes to different positions, when it comes to finding x, y, or z, you would only use the position for the x, y, or z, coordinates, respecitvely, instead of the general one.

For Two masses

x_{COM}=\frac{m_1x_1+m_2x_2}{m_1+m_2}$| The earlier formula, but just defined for two masses, with m for mass, and x being the x position relative to the x direction.</p><div data-type="horizontalRule"><hr></div><p>Canoe trip problem</p><img src="https://assets.knowt.com/user-attachments/08619e2e-353a-4100-b329-7678b9c40536.png" data-width="75$R\theta$| with R for radius, and theta for angle</p><p></p><p>Angular Velocity</p><p></p><p>Def: Rate of change of angle in rad/sec</p><p>Var:$\overrightarrow{w}$</p><p>Unit: rad/s or just s^-1</p><p>Rad is not a real unit</p><p>Equation:$w=\frac{\Delta\theta}{\Delta t}$| with theta for angle, and t for second</p><p></p><p>Connection between linear velocity and angular velocity</p><img src="https://assets.knowt.com/user-attachments/69187fa8-500d-4b07-bd80-cf33eb1993cb.png" data-width="75$\overrightarrow{a}$</p><p>Unit: rad/s²or s^-2</p><p>Equation:$a=\frac{dw}{\differentialD t}=\frac{d^2\theta}{d^2t}$$ | W for angular speed, theta for angle, t for time in seconds, or change in time, time would be the same to divide by, the same for veloocity would be the same in seconds.

Swapping out kinematics equations for angular motion

With this, the theta is in radians here.

Angular velocity depends on angular distance, not regular distance from center.

Inertia