Chapter 10 - Rotation In this chapter we will study the rotational motion of rigid bodies about a fixed axis. To describe this type of motion we will introduce the following new concepts: Angular displacement (symbol: Δθ) Avg. and instantaneous angular velocity (symbol: ω ) Avg. and instantaneous angular acceleration (symbol: α ) Rotational inertia, also known as moment of inertia (symbol I ) The kinetic energy of rotation as well as the rotational inertia; Torque (symbol τ ) We will have to solve problems related to the above mentioned concepts.

The Rotational Variables In this chapter we will study the rotational motion of rigid bodies about fixed axes. A rigid body is defined as one that can rotate with all its parts locked together and without any change of its shape. A fixed axis means that the object rotates about an axis that does not move, also called the axis of rotation or the rotation axis. We can describe the motion of a rigid body rotating about a fixed axis by specifying just one parameter. Every point of the body moves in a circle whose center lies on the axis of rotation, and every point moves through the same angle during a particular time interval.

Consider the rigid body of the figure. We take the z-axis to be the fixed axis of rotation. We define a reference line that is fixed in the rigid body and is perpendicular to the rotational axis. The angular position of this line is the angle of the line relative to a fixed direction, which we take as the zero angular position. In Fig. 11-3, the angular position  is measured relative to the positive direction of the x axis.

From geometry, we know that  is given by (in radians)(10-1) Here s is the length of arc (arc distance) along a circle and between the x axis (the zero angular position) and the reference line; r is the radius of that circle. This angle is measured in radians (rad) rather than in revolutions (rev) or degrees. The radian, being the ratio of two lengths, is a pure number and thus has no dimension. Because the circumference of a circle of radius r is 2  r, there are 2  radians in a complete circle: (10-2) 1 rad = 57,3 0 = 0,159 rev (10-3)

Angular Displacement If the body of Fig rotates about the rotation axis as in Fig. 10-4, changing the angular position of the reference line from  1 to  2, the body undergoes an angular displacement  given by: Δθ = θ 2 – θ 1 (10-4) This definition of angular displacement holds not only for the rigid body as a whole but also for every particle within that body. An angular displacement in the counter- clockwise direction is positive, and one in the clockwise direction is negative.

Angular Velocity Suppose (see Fig ) that our rotating body is at angular position  1 at time t 1 and at angular position  2 at time t 2. We define the average angular velocity of the body in the time interval  t from t 1 to t 2 to be (10-5) Instantaneous angular velocity, ω: (10-6) Unit of ω: rad/s or rev/s

Angular Acceleration If the angular velocity of a rotating body is not constant, then the body has an angular acceleration, α Let  2 and  1 be its angular velocities at times t 2 and t 1, respectively. The average angular acceleration of the rotating body in the interval from t 1 to t 2 is defined as: (10-7) Instantaneous angular acceleration, α: (10-8) Do Sample Problems 10-1 & 10-2, p Unit: rad/s 2 or rev/s 2

A θ s O (10-17) (10-18) (10-19)(10-20)

r O (10-8) (10-22) (10-23)

(10-6)

O riri mimi (10-31) (10-32)(10-33)(10-34) (10-35)

In the table below we list the rotational inertias for some rigid bodies. (10-10)

(10-36)

A

Do Sample Problems 10-6 to 10-8, p

(10-13) (10-39) (10-41) (10-40) Draaimoment

(10-45)

O i riri (10-15) (10-42)

(10-18)