Rotation

Rotational Motion

Angular Velocity/ Linear Velocity/ Angular Acceleration
The Greek symbols ω (angular velocity) is related to velocity and Δθ is related to distance in the kinematics formula velocity= distance/time When the values are substituted in, you get ω=Δθ/Δt. With a constant angular velocity, the angular velocity will be the same at any distance away (r) from the axis of rotation. This value can be related to linear velocity by using velocity=rω. At different points away (r), the linear velocity changes. Angular acceleration (α) can be related to acceleration in the standard kinematic equations. If you plug in ω for velocity, α for acceleration, and Δθ for change in distance, you get the uniformly accelerated rotational motion and constant angular velocity rotational motion.

ω=ω+αt θ=ωt+1/2αt^2 ω^2=ω^2+2αθ

Tangential Acceleration/ Radial Acceleration
The tangential acceleration is shown on a circle as a tangent line. The equation for tangential acceleration is atan = r α. The radial acceleration is ω^2 r. The radial acceleration is toward the center of the circle and keeps the object moving in the circle.

= Axis of Rotation/ Rolling motion = In order for an object to roll instead of slide, their must be friction present. The movement of all particles of an object of any rigid shape around a line called the axis of rotation. If more mass is of an object is closer to the axis of rotation, then the moment of Inertia is smaller.



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Take for instance an a solid cylinder mass of 5 kg with a length of .1 meters and a Annular cylinder (forming or shaped like a ring) with the same mass, with a R1 of .009 meters, R2 of .001 meters. Now place the objects side by side on a ramp (30 degree incline) and release the objects. Which object reaches the end of the ramp first? The solid cylinder will even though both objects have the same mass. The difference is that the solid cylinder has a moment of inertia of .05 while the annular ring has a moment of inertia of 9.05 x 10^-4.

= Torque = The forces in rotational motion are comparable to the forces in linear kinematics. A force perpendicular that is applied to an object a distance away from the axis of rotation causes rotational motion. The distance from the axis of rotation is known as a lever arm (r). The equation then becomes **τ = Fr** Forces are additive when applied in the same direction (clockwise and clockwise) and subtractive when applied in opposite directions (clockwise and counterclockwise). Counter clockwise is positive and clockwise is negative. The summations of torques is similar to the equations F = ma. The F is torque, m is moment of inertia, and a is angular accelerations. The equation then becomes **τ = I** α.

= Rotational Kinetic Energy = The rotational kinetic energy is proportional to the angular velocity squared times the moment of inertia. Therefore Ke(rotational)=1/2[I(ω^2)]. The kinetic energy of system that includes translational kinetic energy and rotational kinetic energy is equal to the translation kinetic plus the rotational kinetic. Pay attention to the friction because it is not kinetic friction but static friction and does not figure into this equation. Work performed by the torque over an angle is given by W= **τ** Δθ.

= Conservation of Angular Momentum = Angular momentum (L) is conserved and very similar to linear momentum. L for a fixed body rotting rotating around the axis of rotation is given by L=I ω. The units for angular momentum kg*m^2/s. The rate of change of change of angular momentum is given by **τ=I**α=ΔL /Δt.

= Right Hand Rule/ Equilibrium = The right hand rule is a convention where if you wrap you fingers in the direction of the motion, the direction the thumb points is the angular momentum. If an object is in equilibrium, then no net torques are applied to the system.

Here is a great website if more explanation is needed [|Rotation] Here is a great video if torque is your weakness.[|Torque] 1d kinematics Energy