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Physics 221, March 1 Key Concepts: •Rotations •Kinematics and dynamics •Energy and angular momentum •Equilibrium Please do not use social media during class. Rotations Extended object can have translational and rotational motion. How do we describe rotational motion? Angular speed: ω = ∆θ/∆t Angular velocity: ω = ∆θ/∆t n, n = direction indicator Angular acceleration: α = ∆ω/∆t Right-hand rule: direction of angular velocity The direction of the angular acceleration is the direction of the change of the angular velocity. A pocket watch and Big Ben are both keeping perfect time. Which minute hand has the larger angular velocity? 1. Big Ben’s 2. The pocket watch’s 3. They both have the same ω. 0% 1 0% 2 0% 3 30 Kinematics and dynamics Motion with constant angular acceleration Kinematic equations: ωf = ωi + α(tf - ti) θf = θi + ωi(tf - ti)+ ½ α(tf - ti)2 ωavg = (ωi + ωf )/2. ωf 2 = ωi 2 + 2 α(θf - θ i). A fan rotating with an initial angular velocity of 1000 rev/min is switched off. In 2 seconds, the angular velocity decreases to 200 rev/min. Assuming the angular acceleration is constant, how many revolutions does the blade undergo during this time? Think about it! How can you approach this problem? A fan rotating with an initial angular velocity of 1000 rev/min is switched off. In 2 seconds, the angular velocity decreases to 200 rev/min. Assuming the angular acceleration is constant, how many revolutions does the blade undergo during this time? Given: wi = 1000 rev/min wf = 200 rev/min, Dw = -800 rev/min a = constant, Dt = 2s Asked: Dq = ? Possible approach to the solution: Find wavg = (wi + wf)/2, then use Dq = wavgDt. Another approach to the solution: Find a = (wf - wi)/ Dt, then use Dq = ωiDt + ½αDt2. Extra credit: A fan rotating with an initial angular velocity of 1000 rev/min is switched off. In 2 seconds, the angular velocity decreases to 200 rev/min. Assuming the angular acceleration is constant, how many revolutions does the blade undergo during this time? 1. 2. 3. 4. 5. 10 20 125 100 1200 0% 1 0% 2 0% 3 0% 4 0% 5 30 Torque A net torque causes angular acceleration. α = τ /I torque = lever arm × force τ=r×F Torque is a vector. It is the cross product of r and F. The cross product of two vectors A and B is defined as the vector C = A×B. The magnitude of C is C = ABsinθ, where θ is the smallest angle between the directions of the vectors A and B. C is perpendicular to both A and B. The direction of C can be found by using the right-hand rule. Let the fingers of your right hand point in the direction of A. Orient the palm of your hand so that, as you curl your fingers, you can sweep them over to point in the direction of B. Your thumb points in the direction of C = A×B. Direction of the torque: Let the fingers of your right hand point from the axis of rotation to the point where the force is applied. Curl them into the direction of F. Your thumb points in the direction of the torque vector. F1, F2 and F3 have the same magnitude. Which statement is correct? 1. 2. 3. 4. All forces produce a torque pointing out of the page. The torque produced by F1 and F3 point in opposite directions. F2 produces no torque, and the torques produced by F1 and F3 have equal magnitudes, F1 produces the torque with the largest magnitude. 0% 1 0% 0% 2 3 0% 4 30 Extra credit: Consider the earth spinning from west to east on its axis. It is slowing down in its rotation due to friction with the air. Using the right hand rule, in what direction is the frictional torque? 1. S pole to N pole 2. N pole to S pole 3. Counterclockwise along the equator 4. Clockwise along the equator 0% 1 0% 2 0% 3 0% 4 30 You are staring at a non-rotating wind turbine and all of a sudden the wind comes up and makes the turbine start rotating clockwise. The net force that causes translational motion is ______ and the torque causing the turbine to rotate is ________. 1. 2. 3. 4. Positive; towards you Positive; towards the right zero; to the left. zero; away from you. 0% 1 0% 2 0% 3 0% 4 30 Moment of inertia The moment of inertia is defined with respect to an axis of rotation. The farther the bulk of the mass is from the axis of rotation, the Greater is the moment of inertia of the object. A cheerleader is given two batons: one gold and one plastic. He is told that both batons have the same mass. He tries spinning each one and finds that the plastic one is much easier to start and stop spinning. He correctly concludes that the plastic baton, in comparison to the gold baton 1. has has more of its mass distributed towards the center. 2. has more of its mass distributed towards the ends. 3. must weigh more. 4. must weigh less. 0% 1 0% 2 0% 3 30 0% 4 Both the disk and the ring have the same mass, and the same radius. Which has the larger moment of inertia? 1. 2. 0% 0% 30 1 2 Energy and angular momentum Energy of an object with translational and rotational motion: translational kinetic energy = ½mvCM2 rotational kinetic energy = ½Iω2 total kinetic energy = ½mvCM2 + ½Iω2 Rolling: vCM = rω total kinetic energy = ½ (m + I/r2)vCM2 Ratio Etrans/Erot = mr2/I Angular momentum L = Iω, ΔL/Δt = Iα = τ, ΔL = τΔt If no external torque acts on a system of interacting objects, then their total angular momentum is constant. Assume a disk and a ring with the same radius roll down an incline of height h and angle theta. If they both start from rest at t = 0, which one will reach the bottom first? 1. The disk 2. The ring 3. They both will arrive at the same time. 0% 1 0% 2 0% 3 30 Extra credit: Suppose the earth’s polar ice caps melted, sending water towards the equator and increasing the moment of inertia of the earth to 1.1 times its present value. The angular velocity of the earth would be 1. 2. 3. 4. 0.82 times its present value. 0.909 times its present value. Unchanged. 1.1 times its present value. 0% 1 0% 2 0% 3 0% 4 30 Equilibrium Mechanical equilibrium: the net external force and the net external torque are zero. Ftot = 0, τtot = 0. Mechanical equilibrium can be stable or unstable. When an object in stable equilibrium is disturbed be a small amount, its total potential energy increases. You can balance a broom on your hand if you’re careful. To do this trick well, you must look at 1. your hand. 2. the top of the broom. 3. the broom’s center of gravity. 4. the middle of the broom. 0% 1 0% 2 0% 3 0% 4 30 You can balance a broom on your hand if you’re careful. To do this trick well, you must look at (3) the broom’s center of gravity. The center of gravity has to stay above the support. You track the center of gravity with your eyes and move your hand accordingly.