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R. Field 10/22/2013 University of Florida PHY 2053Page 1 Angular Momentum The vector angular momentum of the point mass m about the point P is given by: The position vector of the mass m relative to the point P is: (units = kg ∙ m 2 /s) The momentum vector of the mass m is: The magnitude of the angular momentum is: The components of the angular momentum are: Distance from the Point P to the mass m times the perpendicular component of the momentum.

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R. Field 10/22/2013 University of Florida PHY 2053Page 2 Torque The torque vector about the point P due to the force F acting at r is given by: The position vector of the mass m relative to the point P is: (units = N ∙ m) The force acting on the mass m is: The magnitude of the torque is: The components of the torque are: Distance from the Point P to the mass m times the perpendicular component of the force.

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R. Field 10/22/2013 University of Florida PHY 2053Page 3 Rotation: Angular Variables The arc length s is related to the angle (in radians = rad) as follows : Arc Length: Angular Displacement and Angular Velocity: Tangential Velocity and Angular Velocity: (radians/s 2 ) Angular Acceleration: (360 o = 2 rad) Tangential Velocity (radians/second)

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R. Field 10/22/2013 University of Florida PHY 2053Page 4 Rolling Without Slipping: Rotation & Translation If a cylinder of radius R rolls without slipping along the x-axis then: Translational Speed Rotational Speed

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R. Field 10/22/2013 University of Florida PHY 2053Page 5 Translation vs Rotation Translation: Mass: m Rotation: Moment of Inertia: I Force: Torque: Position: x Angular Position: Velocity: v x Angular Velocity: Acceleration: a x Angular Acceleration: If then If constant then constant Momentum Conservation! Angular Momentum Conservation!

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R. Field 10/22/2013 University of Florida PHY 2053Page 6 Exam 2 Fall 2010: Problem 11 A non-uniform cylinder with mass M and radius R rolls without sliding along the floor. If its translational kinetic energy is three times greater than its rotational kinetic energy about the rotation axis through its center of mass (i.e. the central axis of the cylinder), what is its moment of inertia about the central axis? Answer: MR 2 /3 % Right: 44%

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R. Field 10/22/2013 University of Florida PHY 2053Page 7 Example: Rolling without Slipping If a cylinder with moment of inertia I and radius R starts from rest at a height h above the ground and rolls without slipping down an incline. What is its translational speed when it reaches the ground? Example: I = MR 2 /2 (solid cylinder), h = 9.8 m then

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R. Field 10/22/2013 University of Florida PHY 2053Page 8 Example: Rolling without Slipping If a cylinder with moment of inertia I and radius R starts from rest at a height h above the ground and rolls without slipping down an incline. If the cylinder starts from rest at t = 0, when does it reach the ground? Example: I = MR 2 /2 (solid cylinder), h = 9.8 m, = 45 o then

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R. Field 10/22/2013 University of Florida PHY 2053Page 9 Exam 2 Fall 2010: Problem 14 A 100-N uniform plank leans against a frictionless wall as shown. What is the magnitude of the torque (about the point P) applied to the plank by the wall? Answer: 150 N∙m % Right: 42%

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R. Field 10/22/2013 University of Florida PHY 2053Page 10 Exam 2 Fall 2011: Problem 13 A thin stick with mass M, length L, and moment of inertia ML 2 /3 is hinged at its lower end and allowed to fall freely as shown in the figure. If its length L = 2 m and it starts from rest at an angle = 20 o, what is the speed (in m/s) of the free end of the stick when it hits the table? Answer: 7.43 m/s % Right: 14%

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