10 Fixed-Axis Rotation
10 Chapter Review
Key Terms
- angular acceleration
- time rate of change of angular velocity
- angular position
- angle a body has rotated through in a fixed coordinate system
- angular velocity
- time rate of change of angular position
- instantaneous angular acceleration
- derivative of angular velocity with respect to time
- instantaneous angular velocity
- derivative of angular position with respect to time
- kinematics of rotational motion
- describes the relationships among rotation angle, angular velocity, angular acceleration, and time
- lever arm
- perpendicular distance from the line that the force vector lies on to a given axis
- linear mass density
- the mass per unit length λλ of a one dimensional object
- moment of inertia
- rotational mass of rigid bodies that relates to how easy or hard it will be to change the angular velocity of the rotating rigid body
- Newton’s second law for rotation
- sum of the torques on a rotating system equals its moment of inertia times its angular acceleration
- parallel axis
- axis of rotation that is parallel to an axis about which the moment of inertia of an object is known
- parallel-axis theorem
- if the moment of inertia is known for a given axis, it can be found for any axis parallel to it
- rotational dynamics
- analysis of rotational motion using the net torque and moment of inertia to find the angular acceleration
- rotational kinetic energy
- kinetic energy due to the rotation of an object; this is part of its total kinetic energy
- rotational work
- work done on a rigid body due to the sum of the torques integrated over the angle through with the body rotates
- surface mass density
- mass per unit area σσ of a two dimensional object
- torque
- cross product of a force and a lever arm to a given axis
- total linear acceleration
- vector sum of the centripetal acceleration vector and the tangential acceleration vector
- work-energy theorem for rotation
- the total rotational work done on a rigid body is equal to the change in rotational kinetic energy of the body
Key Equations
| Angular position | [latex]\theta =\frac{s}{r}[/latex] |
| Angular velocity | [latex]\omega =\underset{\Delta t\to 0}{\text{lim}}\frac{\Delta \theta }{\Delta t}=\frac{d\theta }{dt}[/latex] |
| Tangential speed | [latex]{v}_{\text{t}}=r\omega[/latex] |
| Angular acceleration | [latex]\alpha =\underset{\Delta t\to 0}{\text{lim}}\frac{\Delta \omega }{\Delta t}=\frac{d\omega }{dt}=\frac{{d}^{2}\theta }{d{t}^{2}}[/latex] |
| Tangential acceleration | [latex]{a}_{\text{t}}=r\alpha[/latex] |
| Average angular velocity | [latex]\overset{–}{\omega }=\frac{{\omega }_{0}+{\omega }_{\text{f}}}{2}[/latex] |
| Angular displacement | [latex]{\theta }_{\text{f}}={\theta }_{0}+\overset{–}{\omega }t[/latex] |
| Angular velocity from constant angular acceleration | [latex]{\omega }_{\text{f}}={\omega }_{0}+\alpha t[/latex] |
| Angular velocity from displacement and constant angular acceleration |
[latex]{\theta }_{