Rotation of Rigid Object
A rigid object is one that is no deformable—that is, the relative locations of all particles of which the object is composed remain constant. All real objects are deformable to some extent; however, our rigid-object model is useful in many situations in which deformation is negligible
Figure 10.1 illustrates an overhead view of a rotating compact disc. The disc is rotating about a fixed axis through O. The axis is perpendicular to the plane of the figure. Let us investigate the motion of only one of the millions of “particles” making up the disc. A particle at P is at a fixed distance r from the origin and rotates about it in a circle of radius r. (In fact, every particle on the disc undergoes circular motion about O.) It is convenient to represent the position of P with its polar coordinates (r, Ɵ), where r is the distance from the origin to P and Ɵ is measured counterclockwise from some reference line as shown in Figure 10.1a. In this representation, the only coordinate for the particle that changes in time is the angle !; r remains constant. As the particle moves along the circle from the reference line (Ɵ= 0), it moves through an arc of length s, as in Figure 10.1b. The arc length s is related to the angle Ɵ through the relationship
Note the dimensions of Ɵ in Equation 10.1b. Because Ɵ is the ratio of an arc lengthand the radius of the circle, it is a pure number. However, we commonly give ! the artificial unit radian (rad), where one radian is the angle subtended by an arc lenght equal to the radius of the arc.
Because the disc in Figure 10.1 is a rigid object, as the particle moves along the circlefrom the reference line, every other particle on the object rotates through the same angle Ɵ Thus, we can associate the angle Ɵ with the entire rigid object as well as with an individual particle. This allows us to define the angular position of a rigid object in its rotational motion. We choose a reference line on the object, such as a line connecting O and a chosen particle on the object. The angular position of the rigid object is the angle Ɵ between this reference line on the object and the fixed reference line in space, which is often chosen as the x axis. This is similar to the way we identify the position of an object in translational motion—the distance x between the object and the reference position, which is the origin, x =0
This quantity is defined as the angular displacement of the rigid object:
The rate at which this angular displacement occurs can vary. If the rigid object spins rapidly, this displacement can occur in a short time interval. If it rotates slowly, this displacement occurs in a longer time interval. These different rotation rates can be quantified by introducing angular speed. We define the average angular speed ω (Greek omega) as the ratio of the angular displacement of a rigid object to the time interval ∆t during which the displacement occurs:
In analogy to linear speed, the instantaneous angular speed & is defined as the limit of the ratio ∆θ / ∆t approaches zero:
Angular speed has units of radians per second (rad/s), which can be written as second-1 (s-1) because radians are not dimensional. We take & to be positive when θ is increasing.
If the instantaneous angular speed of an object changes from &i to &f in the time interval ∆t the object has an angular acceleration. The average angular acceleration α (Greek alpha) of a rotating rigid object is defined as the ratio of the change in the angular speed to the time interval ∆t during which the change in the angular speed occurs:
In analogy to linear acceleration, the instantaneous angular acceleration is defined as the limit of the ratio ∆ω / ∆t as ∆t approaches zero:
Angular acceleration has units of radians per second squared (rad/s2), or just second-2 (s-2). Note that ( is positive when a rigid object rotating counterclockwise is speeding up or when a rigid object rotating clockwise is slowing down during some time interval.
When a rigid object is rotating about a fixed axis, every particle on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration. That is, the quantities θ, ω, & α and characterize the rotational motion of the entire rigid object as well as individual particles in the object. Using these quantities, we can greatly simplify the analysis of rigid-object rotation.
10.2 Rotational Kinematics: Rotational Motion with Constant Angular Acceleration
In our study of linear motion, we found that the simplest form of accelerated motion to analyze is motion under constant linear acceleration. Likewise, for rotational motion about a fixed axis, the simplest accelerated motion to analyze is motion under constant angular acceleration. Therefore, we next develop kinematic relationships for this type of motion.
Notice that these kinematic expressions for rotational motion under constant angular acceleration are of the same mathematical form as those for linear motion under constant linear acceleration. They can be generated from the equations for linear motion by making the substitutions x → θ , v → ω, and a → α. Table 10.1 compares the kinematic equations for rotational and linear motion.
10.3 Angular and Linear Quantities
In this section we derive some useful relationships between the angular speed and acceleration of a rotating rigid object and the linear speed and acceleration of a point in the object. To do so, we must keep in mind that when a rigid object rotates about a fixed axis, as in Figure 10.4, every particle of the object moves in a circle whose center is the axis of rotation.
Because point P in Figure 10.4 moves in a circle, the linear velocity vector v is always tangent to the circular path and hence is called tangential velocity. The magnitude of the tangential velocity of the point P is by definition the tangential speed v = ds/dt, where s is the distance traveled by this point measured along the circular path. Recalling that s = rθ (Eq. 10.1a) and noting that r is constant, we obtain
Because dθ/dt = ω (see Eq. 10.3), we see that
10.4 Rotational kinetic energy
We can derived rotational kinetic energy translation kinetic energy :
cause v=rω, so :
cause mr2 is moments of inertia, so the equation can write: