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Chapter 9 Rotational Dynamics

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**9.1 The Action of Forces and Torques on Rigid Objects**

In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination of translation and rotation.

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**9.1 The Action of Forces and Torques on Rigid Objects**

According to Newton’s second law, a net force causes an object to have an acceleration. What causes an object to have an angular acceleration? TORQUE

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**9.1 The Action of Forces and Torques on Rigid Objects**

The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation.

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**9.1 The Action of Forces and Torques on Rigid Objects**

DEFINITION OF TORQUE Magnitude of Torque = (Magnitude of the force) x (Lever arm) Direction: The torque is positive when the force tends to produce a counterclockwise rotation about the axis. SI Unit of Torque: newton x meter (N·m)

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**9.2 Rigid Objects in Equilibrium**

If a rigid body is in equilibrium, neither its translational motion nor its rotational motion changes.

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**9.2 Rigid Objects in Equilibrium**

EQUILIBRIUM OF A RIGID BODY A rigid body is in equilibrium if it has zero translational acceleration and zero angular acceleration. In equilibrium, the sum of the externally applied forces is zero, and the sum of the externally applied torques is zero.

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**9.2 Rigid Objects in Equilibrium**

Example 3 A Diving Board A woman whose weight is 530 N is poised at the right end of a diving board with length 3.90 m. The board has negligible weight and is supported by a fulcrum 1.40 m away from the left end. Find the forces that the bolt and the fulcrum exert on the board.

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**9.2 Rigid Objects in Equilibrium**

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**9.2 Rigid Objects in Equilibrium**

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**DEFINITION OF CENTER OF GRAVITY The center of gravity of a rigid **

body is the point at which its weight can be considered to act when the torque due to the weight is being calculated.

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9.3 Center of Gravity

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**9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis**

Moment of Inertia, I

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**9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis**

Net external torque Moment of inertia

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**9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis**

ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS Requirement: Angular acceleration must be expressed in radians/s2.

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**9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis**

Example 9 The Moment of Inertia Depends on Where the Axis Is. Two particles each have mass and are fixed at the ends of a thin rigid rod. The length of the rod is L. Find the moment of inertia when this object rotates relative to an axis that is perpendicular to the rod at (a) one end and (b) the center.

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**9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis**

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**9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis**

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**9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis**

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**9.5 Rotational Work and Energy**

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**9.5 Rotational Work and Energy**

DEFINITION OF ROTATIONAL WORK The rotational work done by a constant torque in turning an object through an angle is Requirement: The angle must be expressed in radians. SI Unit of Rotational Work: joule (J)

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**9.5 Rotational Work and Energy**

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**9.5 Rotational Work and Energy**

DEFINITION OF ROTATIONAL KINETIC ENERGY The rotational kinetic energy of a rigid rotating object is Requirement: The angular speed must be expressed in rad/s. SI Unit of Rotational Kinetic Energy: joule (J)

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**DEFINITION OF ANGULAR MOMENTUM **

The angular momentum L of a body rotating about a fixed axis is the product of the body’s moment of inertia and its angular velocity with respect to that axis: Requirement: The angular speed must be expressed in rad/s. SI Unit of Angular Momentum: kg·m2/s

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**PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM **

The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero.

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**Conceptual Example 14 A Spinning Skater **

9.6 Angular Momentum Conceptual Example 14 A Spinning Skater An ice skater is spinning with both arms and a leg outstretched. She pulls her arms and leg inward and her spinning motion changes dramatically. Use the principle of conservation of angular momentum to explain how and why her spinning motion changes.

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Problems to be solved 9.6, 9.12, 9.14, 9.21, 9.25, 9.40, 9.51, 9.61, 9.69, 9.74 B9.1 A bicycle wheel has a mass of 2kg and a radius of 0.35m. What is its moment of inertia? Ans: 0.245kgm2

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**B9. 2 A grinding wheel, a disk of uniform thickness, has a radius of 0**

B9.2 A grinding wheel, a disk of uniform thickness, has a radius of 0.08m and a mass of 2kg. (a) What is its moment of inertia? (b) How large a torque is needed to accelerate it from rest to 120rad/s in 8s? Ans: (a) kgm2 (b) 0.096Nm

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B9.3 A student holding a rod by the centre subjects it to a torque of 1.4Nm about an axis perpendicular to the rod, turning it through 1.3rad in 0.75s. When the student holds the rod by the end applies the same torque to the rod, through how many radians will the rod turn in 1.0s? Ans: 0.58rad

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**B9. 4 An ice skater starts spinning at a rate of 1**

B9.4 An ice skater starts spinning at a rate of 1.5rev/s with arms extended. She then pulls her arms in close to her body, resulting in a decrease of her moment of inertia to three quarters of the initial value. What is the skater’s final angular velocity? Ans: 2rev/s

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