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, 21mv2
, we can convert this formula to rotational kinetic energy question using the rotational motion analogues:
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);
⇒ rotational KE=21Iω2
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.
M: total mass of the object
ICM: moment of inertia about the axis through center of mass
v: translational speed
ω: angular speed
In the previous section, we used the rotational analogue to translate the translational kinetic energy to rotational kinetic energy.
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.m2/s
Newton’s 2nd Law for Rotation
The Newton’s 2nd law also can be written in terms of rotational analogues;
∑F=ma=ΔtmΔv=ΔtΔp, in translational motion “Force” causes linear acceleration
Similarly for rotational motion , ∑=ΔtΔL
∑τ=ΔtΔL=ΔtIΔω=I∝, in rotational motion “Torque” causes rotational acceleration
∑F=ma Newton’s 2nd Law in Translational Motion
∑τ=I∝ Newton’s 2nd Law in Rotational Motion
The law of Conservation of Angular Momentum
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.”
Example: A skater doing a spin on ice, illustrating conservation of angular momentum.
Open arms: Li=Iωi=mri2ωi Closed arms: Lf=Iωf=mrf2ωf
When the arms of the skater are tucked in, the mass is not changing, but the radius of rotation decreases; rf < ri, therefore; mrf2 < mri2
According to the law of conservation of angular momentum; Li=Lf
⇒ωf > ωi “the result would be higher angular velocity”
mrf2 < mri2
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.
A 2.6kg uniform cylindrical grinding wheel of radius 16cm makes 1600rpm.
Rotational kinetic energy and angular momentum
Don't just watch, practice makes perfect.
We have over NaN practice questions in Physics for you to master.