Lecture23-2A

Lecture23-2A - Chapter 13 Chapter 13 Rotational Motion...

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Unformatted text preview: Chapter 13 Chapter 13 Rotational Motion Rotational Motion Chapter 14 Chapter 14 Static Equilibrium Static Equilibrium 3D Angular Momentum: Precession 3D Angular Momentum: Precession What is the rate of precession for the gyro? First we determine the torques and the angular momentum, then apply t = dL/dt . Consider the figure below: Assume that the rotational axis is massless. Then the torque due to gravity is: D m g mgD sin A small change in the angular momentum, dL , is: dL L sin d 3D Angular Momentum: Precession 3D Angular Momentum: Precession What is the rate of precession for the gyro? First we determine the torques and the angular momentum, then apply t = dL/dt . Solving for the precession rate, W = d f /dt : dL dt mgD sin dt L sin d The precession rate is independent of q ! d dt mgD L Example: Precession Example: Precession Initially a gyroscope is spinning with angular speed w and is perfectly balanced so that it is not precessing. When a mass m is hung from the frame the gyro precesses about the vertical axis at a rate W . Find the rotational inertia of the gyro. The torque due to gravity is: The torque is the rate of change of the angular momentum. Hence: Solving for I : mgR mgR dL dt L I I mgR Example: Rolling with Friction Example: Rolling with Friction A solid sphere of mass M and radius R is spinning with angular velocity w o about a horizontal axis. It is dropped vertically onto a surface with a coefficient of kinetic friction m k . (a) Find the expression for the final angular velocity once its achieved pure rolling motion. There are two things happening in this scenario. First the sphere is accelerating in the x direction. It is the frictional force inducing this linear acceleration: The frictional force (via torque) is also responsible for slowing the angular rate: F f k Mg Ma a k g v k gt k MgR I k MgR / 2 MR 2 /5 5 k g /2 R o t o 5 k gt /2 R Example: Rolling with Friction...
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Lecture23-2A - Chapter 13 Chapter 13 Rotational Motion...

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