phys124s11-l02

phys124s11-l02 - Physics 124, Spring 2011 Lecture 1 Summary...

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1 Physics 124, Spring 2011 Lecture 1 Summary  = I  =  rx F L = I  =  rx p  = d L dt v =  r with vectors: v = x r For reference: proof of parallel axis theorem: I q = I cm + m”r 2 I cm = i m i r i 2 = i m i x i 2 y i 2 For convenience, let x cm = 0, y cm = 0 I q = i m i q  r i 2 = i m i [ x i q x 2 y i q y 2 ] Find I about new point q at q x , q y I q = i m i [ x i 2 2x i q x q x 2 y i 2 2y i q y q y 2 ] I p = i m i [ x i 2 y i 2 q x 2 q y 2 ] = i m i [ r i 2 q 2 ] = I cm mq 2 QED m i x i q x =  m i x i q x E = 1 2 I cm 2 1 2 mv cm 2 = mx cm q x = 0
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2 Physics 124, Spring 2011 Lecture 2 Outline Center of mass / gravity Static equilibrium Elastic properties of matter ANNOUNCEMENT : study groups Poll: February 18 or March 11 off? Today we will solve an NPR Car Talk Puzzler: How do find the ¼ full level of a (sideways) cylindrical gas tank? It is not quite halfway from the middle to the bottom. .. 1/2full 1/4full?
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3 Physics 124, Spring 2011 Center of mass / gravity of extended object y x L x The average position of the mass in an object Easy to determine for simple objects – x cm =L x /2, y cm =L y /2 For more complicated objects, or nonuniform densities, calculate sums or integrals, or use gravity in a measurement to determine position of cm We typically consider objects in a uniform gravitational field, so the center of gravity and the center of mass coincide L y Uniform rectanglar solid
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4 Physics 124, Spring 2011 Center of mass of extended irregular object Sum over elements of object to find cm y x x cm = dx x  r m y cm = dy y  r m or x cm = i x i m i m y cm = i y i m i m For cg: x cg = i x i m i g mg = x cm and similarly y cg = y cm Note: x or y in sums is cm position of that element in the sum If g = g x , then you can see that x cm might not equal x cg etc.
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5 Physics 124, Spring 2011 Center of mass of simple-to-calculate object Consider an L shape made up of 4 blocks each a square with sides of length d. Assume all blocks have the same mass M Treat this as a 2-dimensional problem – ignore width in z y x d d x cm = i x i m i m = 3 d / 2  M  3d / 2  M 4M = 3Md 4M = 3 4 d
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6 Physics 124, Spring 2011 Center of mass of simple-to-calculate object Consider an L shape made up of 4 blocks each a square with sides of length d. Assume all blocks have the same mass M
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phys124s11-l02 - Physics 124, Spring 2011 Lecture 1 Summary...

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