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3-Dimensional Rotation Torque and Time Derivative of Angular Momentum Torque about S is equal to the time 3-Dimensional Rotation: derivative of the angular momentum about S Gyroscopes dL τ ext = S S dt 8.01 W13D2 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 Vectors Time Derivative of a Vector of Constant Length: Circular ˆ Motion Consider a vector Akr=+AAzrˆ where Circular Motion: position vector points radially Ar = A sinφ Az = A cosφ outward, with constant magnitude but changes in direction. The velocity vector points in a tangential A vector can change both direction to the circle with a constant magnitude. The magnitude and direction. acceleration vector points radially inward. rr= r ˆ ˆ ˆ Example: Suppose Akr =+ AA zr dθ ddr θ ˆ v = r does not change magnitude but v ==r θ θ dt dt dt only changes direction then 2 ddv θθ⎛⎞ d arr==−vrˆˆ =− ddA θ dθ θ ⎜⎟ = A θ=ˆˆA sinφ θ dt dt⎝⎠ dt dtr dt dt 1 Toy Gyroscope: Forces and Introduction To Gyroscopic Torque Gravitational force acts at the center Motion 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 ˆ ˆ τSS=×rF,cm gravity =×−=bmgbmg rˆ () k θ The direction of the torque about pivot points into the page in the figure Torque: Magnitude of Direction of Angular Angular Momentum Changes Momentum Changes dθ If the flywheel of the gyroscope If the flywheel is spinning, the Ω≡ is not spinning, gyroscope spin angular momentum about dt starts to fall downward and the the center of mass of the torque about the pivot point S flywheel points along the axle, induces the gyroscope to start radially outward; the torque rotating about an axis pointing causes the spin angular into page. momentum to change its d L d 2φ τ = S direction, with precessional Lrspin = I ω ˆ S bmg = IS 2 cm cm S dt dt angular frequency ddθ Torque induces the magnitude LLspin= spin θˆ cm cm d spin spin dt dt of the angular momentum to LL= Ω=θˆˆI ω Ωθ change. dt cm cm cm S Ω≡dθ / dt 2 Gyroscope: Precession Gyroscopic Approximation Torque about the pivot point Flywheel is spinning with an induces the angular angular velocity momentum to change ωspin = ω rˆ S dLS spin Precessional angular velocity τ = bmg =ΩL S dt cm ˆ Ω=Ω k Precessional angular Total angular velocity frequency of the gyroscope total spin bmg bmg ω =Ω+ω Ω= = Lspin I ω Gyroscopic approximation: the cm cm S angular velocity of precession Ω Newton’s Second Law: center is much less than the component of mass undergoes circular of the spin angular velocity ω S , motion 2 F − mg = 0 F =−mbΩ Ω ωS vertical rad Deflection of a Free Particle Table Problem: Gyroscope by a Small Impulse A gyroscope wheel is at one end of an axle of length l . The axle is pivoted at an angle φ with respect to the horizontal. The wheel is set into If the impulse I << p 1 motion so that it executes uniform the primary effect is to precession. The wheel has mass m rotate p about and moment of inertia Icm about its center of mass . Its spin angular the x axis by a small velocity is ωs . Neglect the mass of angle θ . the shaft. What is the precessional frequency of the gyroscope? Which direction does the gyroscope rotate? 3 Deflection of a Free Particle Deflection of a Free Particle by a Small Impulse by a Small Impulse IpF=Δ =ave Δt As a result, L rotates about Δ=LrFτ Δ=×tt Δ the x axis by a small angle ave( ave ) θ. Note that although I is Δ=×LrF Δt ave in the z direction, Δ L is in Δ=×LrI the negative y direction. The application of I causes a change in the angular momentum L through the torque equation. Effect of a Small Impulse on Effect of a Small Impulse on a Tethered Ball a Tethered Ball The ball is attached to a string rotating about a fixed The ball is given an impulse perpendicular to r and to p . point. Neglect gravity. 4 Effect of a Small Impulse on Effect of a Small Impulse on a Tethered Ball a Tethered Ball As a result, L rotates about the x axis by The plane in which the ball moves also rotates about the x axis by the same angle. Note that although I is a small angle θ. Note that although I is in the z direction, Δ L is in the negative y direction. in the z direction, the plane rotates about the x axis. Concept Question: Effect of Effect of a Large Impulse on a Large Impulse on a a Tethered Ball Tethered Ball What impulse I must be given to the ball in order What impulse I must be given to the ball in order to rotate its orbit by 90 degrees as shown without to rotate its orbit by 90 degrees as shown without changing its speed? changing its speed? 5 Solution: Effect of a Large Solution: Effect of a Large Impulse on a Tethered Ball Impulse on a Tethered Ball I must halt the y motion and provide a momentum ΔL cancels the z component of L and adds a component of equal magnitude along the z direction. of the same magnitude in the negative y direction. Effect of a Small Impulse Effect of a Small Impulse Couple on a Baton Couple on a Baton Now we have two equal masses at the ends of a Again note that the impulse couple is applied in the z massless rod which spins about its center. We apply direction. The resulting torque lies along the negative y an impulse couple to insure no motion of the CM. direction and the plane of rotation tilts about the x axis. 6 Effect of a Small Impulse Concept Question: Effect of Couple on Massless Shaft a Small Impulse Couple on of a Baton Massless Shaft of a Baton To make the top of the shaft move in the -y direction Instead of applying the impulse couple to the masses one in which direction should one apply the top half of an could apply it to the shaft to achieve the same result. impulse couple? Solution: Effect of a Small Effect of a Small Impulse Impulse Couple on Couple on Massless Shaft Massless Shaft of a Baton of a Baton The impulse couple Ib applied to the shaft has the Trying to twist the shaft around the y axis causes same effect as the Ia couple applied directly to the the shaft and the plane in which the baton moves masses. Both produce a torque in the - y direction. to rotate about the x axis. 7 Effect of a Small Impulse Effect of a Small Impulse Couple on a Non-Rotating Couple on a Disk Disc The plane of a rotating disk and its shaft behave just This unexpected result is due to the large pre-existing L . like the plane of the rotating baton and its shaft when If the disk is not rotating to begin with, Δ L is also the one attempts to twist the shaft about the y axis. final L . The shaft moves in the direction it is pushed. Effect of a Small Impulse Effect of a Force Couple on Couple on a Disk a Rotating Disk A series of small impulse couples, or equivalently a It does not matter where along the shaft the impulse continuous force couple, causes the tip of the shaft couple is applied, as long as it creates the same torque. to execute circular motion about the x axis. 8 Effect of a Force Couple on Precessing Gyroscope a Rotating Disk ddtLL=Ω dL τ = dt τ = L Ω τ = I ω Ω τ Ω= I ω The precession rate of the shaft is the ratio of the magnitude of the torque to the angular momentum. 9.
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