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Physics Notes:

Rotational Kinematics and Moment of Inertia:

Counter-clockwise is considered the positive direction. Clockwise negative. Avg speed = circumference/time = 2piR/T S, the arc length is equal to RTheta W also = 2pif where f is frequency. 1 revolution is equal to 2pi radians which is equal to 360 degrees The distance traveled DeltaS (linear distance) in a rotating object is equal to rdtheta(in radians) Tangential (linear) speed is v=Rw. So W = v/r Tangential (linear) acceleration a= Ralpha Centripetal acceleration = rw^2 Moment of inertia plays the role of mass in rotational motion. Inertia is equal to the sum of

mass X r^2

KE of rotating system is equal to 1/2IW^2 Moment of inertia depends not only on total mass, but where that mass is located

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Rotation of a rod: Inertia when rotating around the center: 1/12 ML^2 Inertia when rotating around the end: 1/3 ML^2

Parallel Axis Theorem and Torque

KE relative to the center of mass is just = 1/2IcmWcm^2 Ksystem of solid objects: 1/2Mvcm^2+1/2IcmWcm^2 Vcm= Dw Total Inertia = MD^2+Icm. (Parallel Axis theorem) Total inertia is equal to the sum of inertia components Rotational force = Torque (rFtheta) = Ialpha If dW is positive, Alpha points in same direction as W. If negative Alpha points in opposite

direction as W. Torque is the cross product of r and F which is also equal to rFsintheta

Rotational Dynamics

Angular acceleration is equal to 2F/MR Work in Rotations is just FD When a ball is rolling without slipping, Vcm = RW

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KE total of ball that rolls without slipping = 7/10MtotalVcm^2 Vcm of ball that rolls without slipping = sqrt(10/7gH) Angular acceleration when rolling without slipping = a/R

Rotational Statics:

Torque by gravitational force is equal to Rcm X Mg Magnitude of torque by any force = (rsinTheta)F = r(perpendicular)F For statics, Tension will equal Mg/2tantheta Force of hinge in y direction = Mg For rotations around a hinge, angular acceleration is 3g/2L cos(theta). So it is max when theta is

0 and is 0 when theta is -90

In static equilibrium the Acm and Alpha is zero So force acting on the center of mass is zero and the torque acting on the axis is zero

Angular Momentum:

Angular Momentum (L) = r X p The change in angular momentum is equal to the net torque If net torque is zero, then the change in angular momentum is zero, which means that the Total

Angular Momentum is conserved.

Angular momentum of single particle = Iw System of particles having the same angular velocity = ItotalW Angular momentum of solid objects = Iw Lsystem = Lorbit + L spin

Lorbit = Rcm X Mvcm Torque about z axis = Rrad X Fxy Ii/IfWi = Wf Kf = Ii/If Ki Angular Momentum vector always points in the same direction as the angular velocity vector When there is an external torque is not zero, then the angular momentum will change, and this

amount of change is called precession.

Precession rate is the Torque(ext)/L. Units is in radians per second. When we substitute values in this equation we get that the Precession rate is equal to 2gd/R^2w This is valid only if T is small compared to L