Chapter 10. Rotation. 10.1. What is Physics? 10.2. The Rotational Variables 10.3. Are Angular Quantities Vectors? 10.4. Rotation with Constant Angular Acceleration 10.5. Relating the Linear and Angular Variables 10.6. Kinetic Energy of Rotation
10.1. What is Physics?
10.2. The Rotational Variables
10.3. Are Angular Quantities Vectors?
10.4. Rotation with Constant Angular Acceleration
10.5. Relating the Linear and Angular Variables
10.6. Kinetic Energy of Rotation
10.7. Calculating the Rotational Inertia
10.9. Newton's Second Law for Rotation
10.10. Work and Rotational Kinetic Energy
When an object rotates, points on the object, such as A, B, or C, move on circular paths. The centers of the circles form a line that is called the axis of rotation
In this chapter we will examine the rotations of rigid bodies about fixed rotation axes. A fixed axis means the rotation occurs about an axis that does not move
Curl the fingers of your right hand in the direction of the rotation.
Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity.
Average angular acceleration:
SI Unit of Average Angular Acceleration: radian per second squared (rad/s2)
The acceleration is a vector pointing along the axis of rotation. The acceleration vector has the same direction as the change in the angular velocity:
A jet awaiting clearance for takeoff is momentarily stopped on the runway. As seen from the front of one engine, the fan blades are rotating with an angular velocity of –110 rad/s, where the negative sign indicates a clockwise rotation (see Figure). As the plane takes off, the angular velocity of the blades reaches –330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant.
The blades of an electric blender are whirling with an angular velocity of +375 rad/s while the “puree” button is pushed in, as Figure 8.11 shows. When the “blend” button is pressed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of +44.0 rad (seven revolutions). The angular acceleration has a constant value of +1740 rad/s2. Find the final angular velocity of the blades.
If a reference line on a rigid body rotates through an angle θ, a point within the body at a distance r from the rotation axis moves a distance s along a circular arc, where s is given by
The nonuniform circular motion : its tangential speed is changing. An object which is in nonuniform circular motion has both centripetal acceleration ac and a tangential acceleration aT.
Φ= tan-1 (aT/ac)
A helicopter blade has an angular speed of w=6.50 rev/s and an angular acceleration of a=1.30 rev/s2. For points 1 and 2 on the blade in Figure, find the magnitudes of (a) the tangential speeds and (b) the tangential accelerations.
Discus throwers often warm up by standing with both feet flat on the ground and throwing the discus with a twisting motion of their bodies. Figure (a) illustrates a top view of such a warm-up throw. Starting from rest, the thrower accelerates the discus to a final angular velocity of +15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves on a circular arc of radius 0.810 m. Find (a) the magnitude a of the total acceleration of the discus just before it is released and (b) the angle Φ that the total acceleration makes with the radius at this moment.
The rotational kinetic energy KER of a rigid object rotating with an angular speed ω about a fixed axis and having a rotational of inertia I is
Requirement:ω must be expressed in rad/s.
SI Unit of Rotational Kinetic Energy: joule (J)
An object that undergoes combined rotational and translation motion has two types of kinetic energy:
For a continuous body, we define the rotational inertia of the body as
If we know the rotational inertia of a symmetric object rotating about an axis passing through its center of mass, then the rotational inertia I about another parallel axis (where the perpendicular distance between the given axis and the axis through the center of mass is h) is:
Four objects having the same “radius” and mass are shown in the figure that follows. Rank the objects according to the rotational inertia about the axis shown, greatest first.
A thin-walled hollow cylinder (mass=mh, radius=rh) and a solid cylinder (mass=ms, radius=rs) start from rest at the top of an incline (Figure). Both cylinders start at the same vertical height h0. All heights are measured relative to an arbitrarily chosen zero level that passes through the center of mass of a cylinder when it is at the bottom of the incline (see the drawing). Ignoring energy losses due to retarding forces, determine which cylinder has the greatest translational speed upon reaching the bottom.
A net external force causes linear motion to change, but what causes rotational motion to change?
Note: for fixed rotational axis motion, only components of and in the
x-y plane will contribute to the torque along the rotational axis
Figure (a) shows the ankle joint and the Achilles tendon attached to the heel at point P. The tendon exerts a force of magnitude F=720 N, as Figure (b) indicates. Determine the torque (magnitude and direction) of this force about the ankle joint, which is located 3.6×10–2 m away from point P.
ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS:
Requirement: a must be expressed in rad/s2.
are vector components along the same axis
The motor in an electric saw brings the circular blade from rest up to the rated angular velocity of 80.0 rev/s in 240.0 rev. One type of blade has a moment of inertia of 1.41×10–2 kg · m2. What net torque (assumed constant) must the motor apply to the blade?
Figure a shows a uniform disk, with mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration of the falling block, the angular acceleration of the disk, and the tension in the cord. The cord does not slip, and there is no friction at the axle.
The rotational work WR done by a constant torque in turning an object through an angle θis:
For variable torque, rotational work is:
Requirement:θ must be expressed in radians. Unit of work: joule (J)
In addition, we can find the power P associated with the rotational motion of a rigid object about a fixed axis using the equation dW = d :
Note: The signs of both torque and rotational velocity are determined by the right-hand rule.
A rigid sculpture consists of a thin hoop (of mass m and radius R = 0.15 m) and a thin radial rod (of mass m and length L = 2.0 R), arranged as shown in Fig. The sculpture can pivot around a horizontal axis in the plane of the hoop, passing through its center. (a) In terms of m and R, what is the sculpture's rotational inertia I about the rotation axis? (b) Starting from rest, the sculpture rotates around the rotation axis from the initial upright orientation of Fig. What is its rotational speed ω about the axis when it is inverted?
A tall, cylindrical chimney will fall over when its base is ruptured. Treat the chimney as a thin rod of length L = 55.0 m (Fig. 10-20a). At the instant it makes an angle of θ=35o with the vertical, what is its angular speed ωf ?