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November 13, 2014 Teacher note: do torque first - in torque folder Demonstrations · · torque and moment of inertia using meter stick with masses · gyroscope - angular momentum - why doesn't it tip? · bicycle wheel - angular momentum and torque Angular Motion Introduction exploratory · Angular motion Phet Torque Exploratory · Phet I stands for Moment of Inertia τ = Iα Torque is the rotational equivalent to the linear force. Torque causes a rotational or angular acceleration November 13, 2014 November 13, 2014 What is the moment of inertia around a new axis? How are these two things different? Example 8.9, Exercise 8.9 What is the moment of inertia around a new axis? November 13, 2014 Net Torque causes an object to angularly accelerate Net Torque causes an object to angularly accelerate ** In an equation, if the symbol is for a rotational quantity, it must be in radians. Sometimes radians "just appears" or simply "disappears" do not be alarmed by this. This is due to the definition of radians. Which would win a race down an inclined plane, both starting from rest? Which is traveling the fastest? Explain? November 13, 2014 Force causing a torque on the object Force causing a torque on the object radius radius Fg(perp) Fg(perp)*r = τ = I*α Fg(perp) Fg Fg Fg(parallel) Fg(parallel) 2 mg*cosθ *r = ∑mr *α Notice the m's cancel - so the mass of the object is irrelevant If at= α*r Then, while on the ramp we could find the tangential acceleration The angular velocity will be greater if the _______________ is smaller? The angular velocity will be greater if the _______________ is smaller? If the ball gets to the bottom of the ramp and does not accelerate any more, could we find the speed of the ball? If the object gets to the bottom of the ramp and does not accelerate any more, could we find the speed of the object? ** for a hoop or cylindrical shell PEg = KE linear + KE rotational mgh = 1/2mv2 + 1/2Iω2 14 meters tall vt = ωr so ω = vt/r and I = ∑mr2 mgh = 1/2mvt2 + 1/2*∑mr2*(vt / r )2 What cancels? At the bottom there is no net force on the object What kind of energy does it have? be careful Where does it come from? What would the energy conservation equation look like? Hoop Find speed of hoop at the bottom of a 14m high ramp? Find speed of cylinder, sphere check answers try problem 8.34 November 13, 2014 8.34 8.34 Fchain = force of chain causing forward motion Fchain = force of chain causing forward motion Fr = resistive force slowing the wheel down Fr = resistive force slowing the wheel down Both forces cause a torque on the wheel. Both forces cause a torque on the wheel. What is the net torque? What is the net torque? Fchain * write the net torque equation Fchain τchain - τr = Iα Fr Fr 8.34 **Why is one subtracted? it causes a torque in the other direction 8.34 τchain - τr = Iα τchain - τr = Iα (Fchain*.045m) - (120N*.32m) = Iwheel*4.5 rad/s2 (Fchain*.045m) - (120N*.32m) = Iwheel*4.5 rad/s2 * .045m and .32m, why? book gave us diameters, we want radii Fchain * .045m and .32m, why? book gave us diameters, we want radii Fchain *assume wheel is a hoop Fr Iwheel = MR2 = 1.8kg*(.32m)2 = .184 kgm2 *chart page 241 substitute and solve for Fchain *assume wheel is a hoop Fr Iwheel = MR2 = 1.8kg*(.32m)2 = .184 kgm2 substitute and solve for Fchain a. 872 N b. 1400 N November 13, 2014 AP 8.36 8.40 solve first with forces then with energy spool is a solid disk use table on page 241 moment of inertia **so that the accelerations are in the same direction - + at = r*α November 13, 2014 try 8.31, 8.33, 8.36 Conservation of angular momentum and Energy concepts November 13, 2014 Conservation of Angular Momentum, applying ratios length (L) meters vt (m/s) If the objects are moved to half their distance from the axis of rotation, what happens to the following? angular momentum? angular speed? tangential speed? rotational kinetic energy? try 8.48 and 8.49 length = 1.0 meters vt = 5.0 m/s L =? length = 1.0 meters vt = 5.0 m/s L =? If the objects are moved to half their distance from the axis of rotation, what happens to its angular momentum? angular speed? tangential speed? If the objects are moved to half their distance from the axis of rotation, what happens to its angular momentum? angular speed? tangential speed? vt = rω ; ω = 5.00m/s / .5m = 10 rad/s vt = rω ; ω = 5.00m/s / .5m = 10 rad/s L (angular momentum) remains the same due to conservation of mometum Ii = 4kg*(.5m)2 + 3kg*(.5m)2 = 1.75 kgm2 L (angular momentum) remains the same due to conservation of mometum Iω =I ω i i f f Iω =I ω i i f f 1.75kgm2*10 rad/s = .438kgm2*ωf 1.75kgm2*10 rad/s = .438kgm2*ωf ωf = 40 rad /s vt = rω ; .25m * 40 rad/s = 10 m/s Ii = 4kg*(.5m)2 + 3kg*(.5m)2 = 1.75 kgm2 ωf = 40 rad /s vt = rω ; .25m * 40 rad/s = 10 m/s If = 4kg*(.25m)2 + 3kg*(.25m)2 = .438 kgm2 half of the radius --- _____* the angular velocity If = 4kg*(.25m)2 + 3kg*(.25m)2 = .438 kgm2 half of the radius ---4* the angular velocity half of the radius -- ____ the moment of Inertia half of the radius --- _____* the tangential speed half of the radius -- 1/4 the moment of Inertia half of the radius ---2* the tangential speed KE's ? November 13, 2014 What is the form or forms of energy, while a rolling along horizontally? *be careful What is the form or forms of energy, while a rolling object is a rest on top of a hill? zero level zero level Apply conservation of energy formula AP 8.40 Conservation of Energy Example todo Write the conservation of energy formula for the system practice ratio problems - in general - see universal G M1 M1 at rest Ratios with Kepler's third law M2 moment before the object hits the ground height M2 Ground November 13, 2014