Chapter 10 - Rotation In this chapter we will study the rotational motion of rigid bodies about a...
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Transcript of Chapter 10 - Rotation In this chapter we will study the rotational motion of rigid bodies about a...
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- 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: ) Avg. and instantaneous angular velocity (symbol: ) Avg. and instantaneous angular acceleration (symbol: ) Rotational inertia, also known as moment of inertia (symbol I ) The kinetic energy of rotation as well as the rotational inertia; Torque (symbol ) We will have to solve problems related to the above mentioned concepts.
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- 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.
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- 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.
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- 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)
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- Angular Displacement If the body of Fig. 10-3 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: = 2 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.
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- Angular Velocity Suppose (see Fig. 10-4 ) 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, : (10-6) Unit of : rad/s or rev/s
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- Angular Acceleration If the angular velocity of a rotating body is not constant, then the body has an angular acceleration, 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, : (10-8) Do Sample Problems 10-1 & 10-2, p244-246 Unit: rad/s 2 or rev/s 2
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- A s O (10-17) (10-18) (10-19)(10-20)
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- r O (10-8) (10-22) (10-23)
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- O riri mimi (10-31) (10-32)(10-33)(10-34) (10-35)
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- In the table below we list the rotational inertias for some rigid bodies. (10-10)
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- Do Sample Problems 10-6 to 10-8, p254-256
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- (10-13) (10-39) (10-41) (10-40) Draaimoment
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- O 1 2 3 i riri (10-15) (10-42)
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