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Torque

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Correlation between Linear Momentum and Angular Momentum Resistance to change in motion: Linear behavior: Inertia Units M (mass), in kg Angular Rotational Inerita, Symbol I aka Moment of Inertia Has to do with more than just mass, it includes the shape of the object. If x cm is far away, I is bigger. Based on mass and mass distribution.

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Three Important equations for I : Thin Walled Cylinder I = MR 2 (also valid for an orbiting body) Solid Cylinder I = (1/2)MR 2 (easier because the mass is not so far away) Sphere I = (2/5)MR 2 (marble)

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Cause of acceleration Linear Force (Newtons) Angular Torque ( Nm)

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Momentum Linear p = mv ( units kgm/s) Angular L = I (units kgm 2 /s) Bigger mass = harder to stop Faster moving = harder to stop

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Kinetic Energy Linear K = (1/2) mv 2 (Units Joules) Angular K R = (1/2) I 2 A rolling ball has translational and rotational Kinetic energy. Remember: ms become Is, vs become s

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Pivoting Object Take a shape and a piece of paper. Label an x-y axis on the paper. Label pivot point on the shape. Place at origin. Label x cm on shape. Draw weight vector. Draw position vector r from pivot to x cm. Move position vector from origin to x cm.

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Definition: torque Torque is the cause of angular acceleration in the same way that force is the cause of linear acceleration. Symbol (pronounced tao) is the cross product between r and F We say = r cross F = r x F

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Relationship between torque and Newtons second law F = ma rF = rma a is tangential, so a =r rF = torque, = rmr = mr 2 = I ( I=m 1 r 1 2 = m 2 r 2 2 ) So = I ( must be in rad/s 2 )

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Back to picture Draw the angle between the FIRST r vector that starts at the pivot point and the F vector. Then draw the angle between the moved r vector and F. The moved vector is 180-. The old r vector was at . = rF g sin(180-) Mathematically, sin(180-) = sin So = rF g sin() in magnitude. Use RHR for direction.

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On picture, Split weight vector out into components mgcos points toward pivot, mgsin points perpendicular to make it rotate. mgcos gets cancelled out, thats why we use the sin component, = rFsin() That component in this case is spinning it cw, or in the negative direction.

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Lever arm Draw a line from pivot point across to weight vector so that is crosses weight vector at 90 degrees. That line has a magnitude of rsin This is very important. It has a special definition. Its called a lever arm (symbol l ) We can say: = l F

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Cross Product Guided Practice #1 P = 4i + 2j k Q = -3i + 6j -2k P x Q = Answer: 2i + 11j + 30k

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Guided Practice #2 Cross Products P = 2.12i + 8.15j 4.28k N and Q = 2.29i -8.93j 10.5k m Find P x Q Answer: -124i +12.5j 37.6k

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Solution 1) O = 0.17 rad/s (v/r) 2) I = (2/5)MR 2 = 0.0144 kgm 2 3) = 1735 rad/s 4) = use quadratic to find time 5) t = 4.25 s

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Solution Find I I = MR 2 =.625 kgm 2 Find = 16 rad/s Find = = 800 rad = 127 rev

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end https://www.youtube.com/watch?v=GLlpi- 0_lB0 https://www.youtube.com/watch?v=GLlpi- 0_lB0