Rotational kinetic energy and angular momentum  Rotational Motion
Rotational kinetic energy and angular momentum
Lessons
Notes:
In this lesson, we will learn:
 How to calculate rotational kinetic energy?
 Definition of angular momentum
 Newton’s 2^{nd} law for rotation
 The law of conservation of angular momentum
Notes:
 Objects rotating about an axis possess “Rotational Kinetic Energy”
 Objects moving along a straight line possess “Translational Kinetic Energy”.
 Translational kinetic energy is calculated using, $\frac{1}{2}mv^{2}$ , we can convert this formula to rotational kinetic energy question using the rotational motion analogues:
 The center of mass of a rotating object might undergo translational motion (a sphere rolling down an incline), in this case we have to consider both rotational and translational kinetic energy.
If we consider a rotating ball, every point on the ball is rotating with some speed.
The ball is made up many tiny particles, each of mass “$m$”. let’s take “$r$” to be the distance of any one particle from the axis of rotation (O);
$M$: total mass of the object
$I_{CM}$: moment of inertia about the axis through center of mass
$v$: translational speed
$\omega$: angular speed
In like manner, the linear momentum can be changed to angular momentum, using the rotational analogues;
$L$: is the angular momentum with a standard unit of kg.m^{2}/s
The Newton’s 2^{nd} law also can be written in terms of rotational analogues;
$\sum F = \, ma \, =$ $\large \frac{m \Delta v}{\Delta t}$ $=$ $\large \frac{\Delta p}{\Delta t} \quad$, in translational motion “Force” causes linear acceleration
$\qquad \quad$ Similarly for rotational motion , $\sum =$ $\large \frac{\Delta L }{\Delta t}$
$\sum \tau = \,$ $\large \frac{\Delta L}{\Delta t}$ $=$ $\large \frac{I \Delta \omega}{\Delta t}$ $\, = I \propto \quad$, in rotational motion “Torque” causes rotational acceleration
$\sum F = ma \quad$ Newton’s 2^{nd} Law in Translational Motion
$\sum \tau = I \propto \quad$ Newton’s 2^{nd} Law in Rotational Motion
The law of conservation of angular moment states that;
“The total angular momentum of a rotating object remains constant if the net torque acting on it is zero.”
$\sum \tau = 0 \; \Rightarrow \; L_{i} = L_{f} \; \Rightarrow \; I_{i \omega i} = I_{f \omega f}$
Example: A skater doing a spin on ice, illustrating conservation of angular momentum.
Open arms: $\qquad L_{i} = I \omega_{i} = mr^{2}_{i} \omega_{i}$
Closed arms: $\qquad L_{f} = I \omega_{f} = mr^{2}_{f} \omega_{f}$
When the arms of the skater are tucked in, the mass is not changing, but the radius of rotation decreases; $r_{f}$ < $r_{i}$, therefore; $mr^{2}_{f}$ < $mr^{2}_{i}$
According to the law of conservation of angular momentum; $L_{i} = L_{f}$
$L_{i} = L_{f}$
$mr^{2}_{f}$ < $mr^{2}_{i}$

Intro Lesson

2.
A sphere of radius 24.0 cm and mass of 1.60 kg, starts from the rest and rolls without slipping down a 30.0° incline that is 12.0m long.

4.
A 2.6kg uniform cylindrical grinding wheel of radius 16cm makes 1600rpm.