21lecture2a08

# 21lecture2a08 - Physics 2A Lecture 21 Feb 27 Vivek Sharma...

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Physics 2A Lecture 21: Feb 27 Vivek Sharma UCSD Physics

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Moment Of Inertia For Uniform Thin Rod alculate I about axis of rotation thru O, distance h from rod's end Mass uniformly distributed on rod dm dm= M dx M dx LL =⇒ 3 22 2 2 h 1 I= ( 3 3 ) 33 Lh h h MM x xdm xdx M L h ⎡⎤ ⎛⎞ == = ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ ∫∫ 2 0,L MOI if rod rotating thru left (h=0) or right (h=L) end: 1 I = 3 ML
Moment Of Inertia For Thick Hollow Cylinder Pick as volume element a thin cylindrical shell of radius r thickness dr and length L. dV=(2 r)Ldr π 21 dm= (2 r)Ldr. Cylinder thickness = R R ρ 22 11 2 2 2 2 1 3 44 (R R )(R +R (2 r)Ldr 2 L r dr 2L (R R ) ) RR I r ρπ πρ == = −= ∫∫ 2 2 2 2 1 1 2 2 L(R R ) 1 Cylinder vol ume V= L(R R ) I= ( ) 2 M MR R ⇒+ −⇒ = 2 1 1 For a solid cylinder R =0 2 MR

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Cable Unwinding Off A Cylinder grav Cable wrapped around cylinder (mass M) is attached to object mass m. As cable unwinds, U converted to kinetic energy. Find speed of object as it hits floor 11 2 2 22 K 00 UK U mgh mv I ω += + + + 2 1 2 2 1/ 2 v mgh mv MR R gh v Mm ⎛⎞ ⇒= + ⎜⎟ ⎝⎠ + Cylinder is Solid

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Moment of Inertia of Objects About Their Axis Of Rotation
Parallel Axes Theorem A body does not have only one moment of Inertia, its has many! As many as the axes about which it can rotate many Relation between MOI about an axis of rotation thru its center of mass & MOI about any other axis || to C.O.M but displaced by distan e c d axis 2 P I cm IM d =+

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Torque at a point, due to an applied
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21lecture2a08 - Physics 2A Lecture 21 Feb 27 Vivek Sharma...

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