Chapter 12 - Rotation of a Rigid Body

Rotational Energy

A rotating body has kinetic energy. But the speed depends on the positions. Moment of inertia is the rotational equivalent of mass. The mass near the center (smaller values of ri) lowers the moment of inertia. Krot = ½Iω^2 Ug = Mgycm

Moment of Inertia

Describes how resistant an object is to angular acceleration. An object's moment of inertia depends on the axis of rotation. I=∑mr^2 = ∫r^2 dm = ∫x^2+y^2

Moment of Inertia for Rotation About P

Ip = Icm + MR^2

Critical angle

Reached when the center of mass is directly over the pivot point. θc = arctan(t/2h) t= width h=height of center of mass

Ropes and Pulleys Vobj Aobj

Ropes and Pulleys: Vobj = |ω|R Aobj = |α|R

Tangential Velocity (vt) (equation)

ds/dt = r(dθ/dt) vt=ωr

Tangential and Angular Acceleration (At)

dvt/dt = r(dω/dt) = rα

Arc Length

s=rθ

Radial Acceleration (Ar)

v^2/r = ω^2r

Rotational Rolling Energy Krolling

Krolling = 1/2Iω^2 + 1/2Mv^2 = Krot + Kcm In other words, the rolling motion of a rigid body can be described as a translation of the center of mass (with kinetic energy Kcm) plus a rotation about the center of mass (with Kinetic Energy Krot)

Rolling Motion Vbottom Vtop

Rolling Motion Vbottom = 0 Vtop = 2Vcm =2ωr

Angular Acceleration (α)

The angular acceleration is the change of angular velocity divided by the time interval during which the change occurred. α = dω/dt

Gravitational Torque

τ=-Mgxcm

Newtons Laws ---> Rotational Dynamics

τnet = (∑mr^2)α = Iα

Angular Velocity (ω)

The angular velocity is the angle swept out divided by the time it took to sweep out the angular displacement. ω = dθ/dt

Center of Mass

The center of mass is the mass-weighted center of the object. Before you can integrate you must replace dm by an equivalent expression involving a coordinate differential such as dx or dy. dm/M = dx/L or dm/M = da/A

The Parallel-Axis Theorem

The moment of inertia depends on the rotation axis. Suppose it's rotating about the off-center axis. Where the axis of interest is distance d from a parallel axis. I = Icm + md^2

Torque (2 defs)

The rotational equivalent of force. 1. Torque is due to the tangential component of force. τ=rFt 2. The moment arm is the distance between the pivot point and the line of action. The line of action is the line along which the force acts. τ=rFsin(θ) The θ is from the pivot/radial line to the line of action.