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3-Dimensional Rotation: Gyroscopes. 8.01 W13D2. Torque and Time Derivative of Angular Momentum. Torque about S is equal to the time derivative of the angular momentum about S
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3-Dimensional Rotation: Gyroscopes 8.01 W13D2
Torque and Time Derivative of Angular Momentum Torque about S is equal to the time derivative of the angular momentum about S If the magnitude of the angular momentum is constant then the torque can cause the direction of the angular momentum to change
Time Derivative of a Vector Consider a vector where A vector can change both magnitude and direction. Example: Suppose does not change magnitude but only changes direction then
Time Derivative of Vectors of Constant Length: Circular Motion Circular Motion: position vector points radially outward, with constant magnitude but changes in direction. The velocity vector points in a tangential direction to the circle with a constant magnitude. The acceleration vector points radially inward.
Deflection of a Free Particle by a Small Impulse If the impulse << the primary effect is to rotate about the x axis by a small angle .
Deflection of a Free Particle by a Small Impulse The application of causes a change in the angular momentum through the torque equation.
Deflection of a Free Particle by a Small Impulse As a result, rotates about the x axis by a small angle . Note that although is in the z direction, is in the negative y direction.
Effect of a Small Impulse on a Tethered Ball The ball is attached to a string rotating about a fixed point. Neglect gravity.
Effect of a Small Impulse on a Tethered Ball The ball is given an impulse perpendicular to and to .
Effect of a Small Impulse on a Tethered Ball As a result, rotates about the x axis by a small angle . Note that although is in the z direction, is in the negative y direction.
Effect of a Small Impulse on a Tethered Ball The plane in which the ball moves also rotates about the x axis by the same angle. Note that although is in the z direction, the plane rotates about the x axis.
Concept Question: Effect of a Large Impulse on a Tethered Ball What impulse must be given to the ball in order to rotate its orbit by 90 degrees as shown without changing its speed?
Effect of a Large Impulse on a Tethered Ball What impulse must be given to the ball in order to rotate its orbit by 90 degrees as shown without changing its speed?
Solution: Effect of a Large Impulse on a Tethered Ball must halt the y motion and provide a momentum of equal magnitude along the z direction.
Solution: Effect of a Large Impulse on a Tethered Ball cancels the z component of and adds a component of the same magnitude in the negative y direction.
Effect of a Small Impulse Couple on a Baton Now we have two equal masses at the ends of a massless rod which spins about its center. We apply an impulse couple to insure no motion of the CM.
Effect of a Small Impulse Couple on a Baton Again note that the impulse couple is applied in the z direction. The resulting torque lies along the negative y direction and the plane of rotation tilts about the x axis.
Effect of a Small Impulse Couple on Massless Shaft of a Baton Instead of applying the impulse couple to the masses one could apply it to the shaft to achieve the same result.
Concept Question: Effect of a Small Impulse Couple on Massless Shaft of a Baton To make the top of the shaft move in the -y direction in which direction should one apply the top half of an impulse couple?
Solution: Effect of a Small Impulse Couple on Massless Shaft of a Baton The impulse couple Ib applied to the shaft has the same effect as the Ia couple applied directly to the masses. Both produce a torque in the - y direction.
Effect of a Small Impulse Couple on Massless Shaft of a Baton Trying to twist the shaft around the y axis causes the shaft and the plane in which the baton moves to rotate about the x axis.
Effect of a Small Impulse Couple on a Disk The plane of a rotating disk and its shaft behave just like the plane of the rotating baton and its shaft when one attempts to twist the shaft about the y axis.
Effect of a Small Impulse Couple on a Non-Rotating Disc This unexpected result is due to the large pre-existing . If the disk is not rotating to begin with, is also the final . The shaft moves in the direction it is pushed.
Effect of a Small Impulse Couple on a Disk It does not matter where along the shaft the impulse couple is applied, as long as it creates the same torque.
Effect of a Force Couple on a Rotating Disk A series of small impulse couples, or equivalently a continuous force couple, causes the tip of the shaft to execute circular motion about the x axis.
Effect of a Force Couple on a Rotating Disk The precession rate of the shaft is the ratio of the magnitude of the torque to the angular momentum.
Toy Gyroscope: Forces and Torque Gravitational force acts at the center of the mass and points downward Contact force between the end of the axle and the pylon Torque about the contact point due to gravitational force The direction of the torque about pivot points into the page in the figure
Torque: Magnitude of Angular Momentum Changes If the flywheel of the gyroscope is not spinning, gyroscope starts to fall downward and the torque about the pivot point Sinduces the gyroscope to start rotating about an axis pointing into page. Torque induces the magnitude of the angular momentum to change.
Direction of Angular Momentum Changes If the flywheel is spinning, the spin angular momentum about the center of mass of the flywheel points along the axle, radially outward; the torque causes the spin angular momentum to change its direction, with precessional angular frequency
Gyroscope: Precession Torque about the pivot point induces the angular momentum to change Precessional angular frequency of the gyroscope Newton’s Second Law: center of mass remains at rest
Gyroscopic Approximation Flywheel is spinning with an angular velocity Precessional angular velocity Total angular velocity Gyroscopic approximation: the angular velocity of precession is much less than the component of the spin angular velocity ,
Table Problem: Gyroscope A gyroscope wheel is at one end of an axle of length l . The axle is pivoted at an angle with respect to the horizontal. The wheel is set into motion so that it executes uniform precession. The wheel has mass m and moment of inertia Icm about its center of mass . Its spin angular velocity is s . Neglect the mass of the shaft. What is the precessional frequency of the gyroscope? Which direction does the gyroscope rotate?
