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Rotational motion Today’s Big Two 1) Newton’s law for rotational motion. 2) Rotational and translational kinetic energy.

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Rotational KinematicsConstant α

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Dynamics: Translation/Rotation analogy:

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Angular motion applet

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τ=Iα What forces are related to the following torques : Ball rolling down an incline Hanging mass attached with a string to a pulley Meter-stick pendulum Which of these have constant torques? Linking translation to rotational quantities: ∑τ= ∑F*r=(ma)*r=(mrα)*r=mr 2 α. Moment of inertia I as a measure of a body’s to resist the change in rotation is dependent on the location of the rotation

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Tutorial: Rotational motion

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Experiment 1: Unwinding with gravity (Department of Mathematics and Physics Wake Technical Community College) Introduction: A type of window blind available from home-improvement stores is designed so that the blind rolls up when raised and rolls down on strings. The design needs to be concerned with the strength of the string used to hold the blind as it rolls up and down. The question is whether or not it can withstand maximum tension if say, the blind falls under the influence of gravity. A model will be built and tested as shown below.

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Unwinding with gravity Cylinder with string wrapped around it Tools: Uniform cylinder, string, support rods, stopwatch, timer and photogates,ruler, balance scale, mass set.

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Motion equation for the falling cylinder Draw an extended free body diagram. State the equations for translational and rotational motion. Is the cylinder in equilibrium? What causes the rotation: Force due to gravity? The tension? Calculate the expected (theoretical) acceleration by combining the translational and rotational equations, and the definition of the moment of inertia of the cylinder.

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Report 1)Describe the methods you use to determine the acceleration of the descending cylinder(figures should be drawn. Include a free body diagram showing all the forces on the cylinder) 2)State any additional information that is needed to determine acceleration (physical or mathematical) 3)State what information can be ignored. Justify your reasoning 4)Record all measurements taken. (several measurements and their averages Values should be included) 5)Submit a data table that includes mass, radius and measured accelerations form the cylinder. 6)Submit a calculation table that includes net torque, angular acceleration, moment of inertia and string tension. 7)List your results for the acceleration of the descending cylinder. 8) State how confident you are with your results.

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Analysis 1.Discuss the limitations of your measurements taken. 2.Does the radius of the cylinder have any effect on the rate of acceleration as it falls? 3.Is the cylinder in equilibrium (translational and rotational ) as it falls? 4.Doe the cylinder have a high or low moment of inertia? How is that quantity measured? 5. Determine a way how to calculate the expected acceleration based on the mass of the cylinder. 6.Compare your calculated and experimentally measured acceleration (%error) 7.Does the tension remain constant as the cylinder is falling? If not, when does the tension reach its maximum value? Explain. 8.What is the minimum tension required for your cylinder falling under the influence of gravity? 9. How well dopes this “scale model” represent the window blind discussed in the introduction?

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Hint: Use the work energy theorem in the form W nc = ΔK + ΔU where the kinetic energy is translational and rotational.

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Which object will arrive first at the bottom of the incline?

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Experiment 2: Sphere rolling down an incline without sliding θ H R

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Motion equation for the rolling sphere Draw an extended free body diagram. State the equations for translational and rotational motion. Is the sphere in equilibrium? What causes the rotation? Calculate the expected (theoretical) speed at the bottom of the incline by combining the translational and rotational equations, and the definition of the moment of inertia of the sphere.

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Now use W nc =∆K+∆U to find the theoretical speed Write down the expression for W nc Write down the expression for ∆K Write down the expression for ∆U

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Trajectory motion: Predict the horizontal range x. θ H y x v Given: v, y, θ Find: x

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Trajectory Motion: Equipment Dynamics track Ball Support block Inclinometer Meter stick Plumb bob Carbon paper Target sheet We will let a ball roll down a ramp and project out over the floor. We need only two measurements to predict where the ball will hit the floor: the length and angle of inclination of the track, and the height of the bottom of the ramp above the floor. To make a record of the impacts, we will place carbon-paper target sheets at the predicted impact point.

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θ H R Height of incline: Angle of incline: Height above floor: Distance along the floor: 1. 2. 3. 4. 5. v experiment =v average ±σ V theoretical = f roll = Table 1: Experiment 2

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Lab report Names and roles (1point) Title (max 2 points) Abstract (max 15 points) Introduction (max 5 points) Materials and methods (max 10 points)

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Lab report Reporting the results (max 10 points)

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Lab report Discussion (max 10 points) Conclusion (max 10 points) Data (max 10 points) Presentation (max 10 points) Overall impact (max 10 points)

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Experiment 1: Newton’s law

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Experiment 1: Energy conservation

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Experiment 2: Newton’s law

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Experiment 2: Energy conservation

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