This lesson will show how to find the velocity and acceleration of a particular point in a rotating rigid body. To find the kinetic energy of a rotating body, the equation K = (1/2)mu2 is needed for a particle. So, the velocity, u, for each particle in the body must be known. Therefore, it becomes necessary to develop general relations between the angular velocity and acceleration of a rigid body rotating about a fixed axis and the linear velocity and acceleration of a specific point or particle in the body.
When a rigid body rotates about a fixed axis, any chosen point, or particle in the body moves in a circular path. The circle lies in a plane perpendicular to the axis and is centerd on the axis. The speed of a particle is directly proportional to the body's angular velocity; the faster the body rotates, the greater the speed of each particle. In the following figure, point P is a constant distance r from the axis of rotation, so it moves in a circle of radius r.
At any time the angle θ (in radians) and the arc length s are related by:
s = rθ
The time derivative of this equation is taken, noting that r is constant for any specific particle, and the aboslute value is taken of both sides:
ds
dt
= r
dθ
dt
Now, |ds/dt| is the absolute value of the rate of change of arc length, which is equal to the instantaneous linear speed u of the particle. Analogously, |dθ/dt|, the absolute value of the rate of change of the angle, is the instantaneous angular speed ω — that is, the magnitude of the instantaneous angular velocity in rad/s. Thus:
u = rω
This is the relation between linear and angular speed. The farther a point is from the axis, the greater its linear speed. The direction of the linear velocity vector is tangent to its circular path at each point.
Acceleration
The acceleration of a particle moving in a circle can be represented in terms of its centripital and tangential components, arad and atan, as with the lesson Non-Uniform Circular Motion. It is found that the tangential component of acceleration atan, the component parallel to the instantaneous velocity, acts to change the magnitude of the particle's velocity (i.e., the speed) and is equal to the rate of change of speed. Taking the derivative of Eq1, it is found that:
atan =
du
dt
= r
dω
dt
= rα
This component of a particle's acceleration is always tangent to the circular path of the particle.
The component of the particle's acceleration directed toward the rotation axis, the centripetal component of acceleration arad, is associated with the change of direction of the particle's velocity. In the lesson Non-Uniform Circular Motion, the relation arad = u2/r was developed. This can be expressed in terms of ω by using Eq1.
arad =
u2
r
= ω2r
This is the equation for the centripital acceleration of a point on a rotating body.
This is true at each instant even when ω and u are not constant. The centripetal component always points toward the axis of rotation.
The vector sum of the centripetal and tangential components of acceleration of a particle in a rotating body is the linear acceleration aarrow.
