ROTATIONAL MOTION

           ROTATIONAL MOTION

Brief summary:

Rotational motion is more complicated than linear motion, and only the motion of rigid bodies will be considered here. A rigid body is an object with a mass that holds a rigid shape, such as a phonograph turntable, in contrast to the sun, which is a ball of gas. Many of the equations for the mechanics of rotating objects are similar to the motion equations for linear motion.

Angular velocity and angular acceleration

The angular displacement of a rotating wheel is the angle between the radius at the beginning and the end of a given time interval. The SI units are radians. The average angular velocity

and is measured in radians/second/second or rad/s ².

Consider a wheel rolling without slipping in a straight line. The forward displacement of the wheel is equal to the linear displacement of a point fixed on the rim. 

d = S = rθ

In this case, the average forward speed of the wheel is v = d/ t = ( rθ)/ t = rω, where r is the distance from the center of rotation to the point of the calculated velocity. The direction of the velocity is tangent to the path of the point of rotation.

The average forward acceleration of the wheel is a _T = r(ω f − ω o)/ t = rα. This component of the acceleration is tangential to the point of rotation and represents the changing speed of the object. The direction is the same as the velocity vector.

The radial component of the linear acceleration is a _r = v ²/ r = ω ² r.

Torque

It is easier to open a door by pushing on the edge farthest from the hinges than by pushing in the middle. It is intuitive that the magnitude of the force applied and the distance from the point of application to the hinge affect the tendency of the door to rotate. This physical quantity, torque, is t = r × F sin θ, where F is the force applied, r is the distance from the point of application to the center of the rotation, and θ is the angle from r to F.

Moment of inertia

Substitute Newton's second law into the definition for torque with θ of 90 degrees (a right angle between F and r) and use the relationship between linear acceleration and tangential angular acceleration to obtain t = r F = rma = mr ² ( a/ r) = mr ²α. The quantity mr ² is defined as moment of inertia of a point mass about the center of rotation.

Imagine two objects of the same mass with different distribution of that mass. The first object might be a heavy ring supported by struts on an axle like a flywheel. The second object could have its mass close to the central axis. Even though the masses of the two objects are equal, it is intuitive that the flywheel will be more difficult to push to a high number of revolutions per second because not only the amount of mass but also the distribution of the mass affects the ease in initiating rotation for a rigid body. The general definition of moment of inertia, also called rotational inertia, for a rigid body is I = ∑ m _i r _i ² and is measured in SI units of kilogram‐meters ².