Rotational Mechanics

Sub-Concepts

• Newton’s 2nd law in rotation
• Energy in Rotation
• Pulley Dynamics

Newton’s 2nd Law in Rotation

$I_o{\omega }^{\prime }=M_o$
$\omega ={\theta }^{\prime }=\frac{d\theta }{dt}$
Where…
theta : angular displacement of a mass about an axis through point O $I_o$: mass moment of inertia about point I $I$ = $\int r^{2}dm$

Parallel Axis Theorem

If rotation is about an axis parallel to the axis of symmetry of the object with distance d
$I=I_s+m{d}^2$

Examples

Examples are found within the week 2 section of this notebook

Energy of Rotational Motion

See Energy

Pulley Mechanics

For a pulley with an inertia of I: If $I\cdot {\theta }^{{}^{\prime \prime }}=0$ then either angular acceleration is 0, or I = 0 because mass or radius is negligible.
Then… T_1=T_2.

The cable of the pulley should also be considered inextensible such that
$x_1=x_2;x_1^{\prime }=x_2^{\prime }$
No slip should be assumed between the cable and a pulley such that:
$x_1=R\theta$ and $x_1^{\prime }=R{\theta }^{\prime }$

These are important terms for rotational energy problems.
The equation:
$x_1\left(t\right)=\frac{1}{2}A{t}^2$
Describes the motion of a pulley system assuming the system in inextensible and that no slip occurs.

By rearranging:

$x_1^{\prime }\left(t\right)=At$
$x_1^{\prime 2}\left(t\right)=\frac{1}{2}A{t}^2\left(2A\right)$
$x_1^{\prime 2}\left(t\right)=x_1\left(t\right)\left(2A\right)$
Which when A is subbed in, is the same as energy :)
$x_1^{\prime }=\sqrt{{\frac{2m_1-m_{2}gx1}{m_1+m_2+\frac{I}{R^2}}}_{}}$