Lecture214Week7 - General case of motion In general the...

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05/13/09 Physics 214 Fall 2008 1 General case of motion In general the motion of an object consists of translation and rotation. Translation we have dealt with in straight line motion A wheel is an excellent example of rotation We define an axis and counterclockwise is positive. 57.3 0 v R +
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05/13/09 Physics 214 Fall 2008 2 Rotational Motion Consider an object moving in a circular path. It has velocity, acceleration, kinetic energy and momentum but these are not the simplest variables Displacement we use the angle θ measured in radians Angular velocity ω = Δθ/Δt Angular acceleration α = Δω/Δt 1 revolution/sec = 2π radians/sec Since the time for one revolution is 2πr/v = 2π/ω then v = rω All parts of a rotating wheel have the same ω but The further from the center the bigger is v θ v R
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05/13/09 Physics 214 Fall 2008 3 Constant angular acceleration Apart from changing variables the equations are identical to linear motion with constant acceleration Displacement d θ Velocity v = Δd/Δt ω = Δθ/Δt Acceleration a = Δv/Δt α = Δω/Δt
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05/13/09 Physics 214 Fall 2008 4 Forces and torques If we apply a force to a bicycle wheel that is free to rotate for a given force it is easier to rotate the wheel the further you are from the axle. In the picture shown below each of the single weights on it’s own will cause the rule to rotate but the two together can be balanced. A force applied to an object, in general makes that object rotate and the action of the force we call a TORQUE and Τ = FL where L is the perpendicular distance to the line of action of the force. Once again + is counterclockwise and – is clockwise and the net torque is sum of all torques.
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05/13/09 Physics 214 Fall 2008 5 Using a wrench In our everyday life we are limited in the force we can apply but if we increase the lever arm we can increase the torque. We can turn a very tight nut by applying a force F at a large radius R If the radius of the nut is r then to just move the nut F R F f
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05/13/09 Physics 214 Fall 2008 6 Center Gravity There is a point associated with any body that allows it to be balanced by a single force upward without any rotation. This point is called the Center of Mass It is like the whole mass of the body is concentrated at that point. If we hold an object up on a piece of string the center of mass will lie on the line of the string.
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05/13/09 Physics 214 Fall 2008 7 Walking the plank If we have an object like a plank or a teeter totter that pivots about a single point then it will be exactly balanced if the clockwise torques equal the counterclockwise torque. As the boy walks out on the plank the clockwise torque increases until it becomes larger than the counterclockwise torque and then the plank rotates M plank gd = M child gx d x
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05/13/09 Physics 214 Fall 2008 8 Newtons second Law For the object shown F = ma v = rω and Δv = Δω x r and a = αr F = ma Fr = T = mar = mr 2 α = I α I plays the role of mass for rotation F= mr 2 a
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05/13/09 Physics 214 Fall 2008 9 Moment of Inertia The Moment of Inertia is always of the form I = mass times a length squared and it depends on the distribution of mass about the axis of rotation
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