# Ans the cm of each sphere is at their respective

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ANS: The CM of each sphere is at their respective centers (WHY?). The computation of the CM is then the same as before! Now (in meters) so 0 1m 3m x 2m x CM = m 1 x 1 + m 2 x 2 m 1 + m 2 = (1 . 0)(0) + (2 . 0)(3 . 0) 1 . 0 + 2 . 0 m = 2 . 0 m (and y CM =0) r 1 = (0 , 0) , r 2 = (3 . 0 , 0) 1kg 2kg 0 1m 3m x 2m CM 1kg 2kg The CM is on the line between the centers, 2/3 of the way from 1kg to 2 kg Copying from previous slide:

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Phys 2A - Mechanics Kinetic Energy of Rotation axis of rotation Rigid body rotating about axis with angular velocity ω . 1 2 i N Compute kinetic energy of i-th element: K i = 1 2 Δ m i v 2 i v i is tangential velocity of circular motion about axis of rotation, with radius r i : K i = 1 2 Δ m i ( ω r i ) 2 = 1 2 ω 2 ( Δ m i r 2 i ) r i v i = ω r i Sum over all elements to get total kinetic energy of rotation: K = 1 2 ω 2 N i =1 r 2 i Δ m i = 1 2 ω 2 I where I is the “moment of inertia” I N i =1 r 2 i Δ m i = V r 2 dm
Phys 2A - Mechanics Summary: For a rigid body rotating about a fixed axis the rotational kinetic energy is K = 1 2 I ω 2 where I is the moment of inertia defined by I V r 2 dm For a collection of particles rotating about a fixed common axis of rotation (as in, eg, a dumbbell) The symbol “ r” or “ r i ” stands for the distance from the axis of rotation to the mass element dm, not the distance from an origin of a coordinate system. It is sometimes labeled r to stress this point. I N i =1 r 2 i m i

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Phys 2A - Mechanics Moments of inertia of common shapes (some we will compute later)
Phys 2A - Mechanics Rolling without slipping Rolling without slipping means that the point of contact is (instantaneously) not moving with respect to the floor. Rolling: “ round” object like a ball or disk on a “floor” (fixed surface). (“ round: ” circular symmetry when rotated about axis of rotation) When this happens, the translational velocity of the axis, v 0 , is related to the angular velocity of rotation and the radius by v 0 = ω R

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Phys 2A - Mechanics Two ways to see this: 1. 2. Point of contact P is instantaneous axis of rotation. Axis at A moving with velocity V a distance R away from axis, so me must have A ω = V R V 2 V
Phys 2A - Mechanics Kinetic Energy: CM axis In general case, rigid body can rotate about axis while the axis is moving. When the axis 1. Goes through CM, and 2. Moves parallel to itself (fixed orientation) the total kinetic energy is K = 1 2 MV 2 CM + 1 2 I CM ω 2 velocity of CM of rigid body V CM = where ω = angular velocity of rotation about axis through CM

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Phys 2A - Mechanics
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• Fall '07
• Hicks

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