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Rotational Motion PHYSICS 1 Rotational Advanced Placement Motion Presenter 2014-2015 Rotational Motion What I Absolutely Have to Know to Survive the AP* Exam The rotational kinematic equations are rotational relationships between the angular displacement, angular velocity, angular acceleration, and time that are only true when the angular acceleration is constant (i.e. when the angular acceleration is not a function of θ as one example). There exists an almost perfect parallel between translational and rotational motion. In a system in which there is both rotation and translation, you must include both rotational and translational kinetic energy in the same conservation of energy expressions. Linear and Angular analogs - variables Linear Angular x Linear Distance (m) Rotational distance θ (radians) Δx Linear Displacement (m) Rotational Δθ displacement (radians) v Linear Velocity (m/s) Rotational velocity ω (radians/s) 2 a Linear Acceleration (m/s ) Rotational acceleration α (radians/s2) m Mass (kg) Rotational inertia Ι (kg.m2) . F Force (N) Torque (N m) τ Rotational Inertia for a system of point masses = I = Σmr 2 Rotational Inertia for common objects (not necessary to memorize) 1 Solid Cylinder or Disc I = mr 2 2 Hoop about center axis I = mr 2 2 Solid Sphere I = mr 2 5 Torque Just as a non-zero net force causes a linear acceleration, a non-zero net torque will cause an angular acceleration. A torque can be thought of as a twist, just as a force is a push or pull. It is a torque that affects an object’s angular velocity. Torque is not energy, however and the units are mN or N.m, not Joules. Torque = τ = r F = rF sinθ ⊥ Where AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this material. Copyright © 2013 National Math + Science Initiative®, Inc., Dallas, TX. All rights reserved. Rotation I: Rotational Kinematics & Energy F = force that is being applied to object (N) r = displacement from the point of rotation to the point of force application (m) θ = angle between the force vector and the displacement vector Relationships between the linear and angular variables when an object is rotating around a fixed axis or rolling without slipping. x = rθ v = rω aT = rα v2 a = a = = ω 2r C R r Where r = the radius of the rotating object in meters 2 aT = tangential acceleration in m/s 2 aC = centripetal acceleration (also called radial acceleration) in m/s Key Formulas and Relationships Linear and Angular analogs – kinematics and energy equations Linear Angular Constant Motion x x vt t = o + θ = θo +ω Motion with Constant Acceleration v = vo + at ω = ω o +αt 1 1 x = (v + v)t θ = (ω +ω )t 2 o 2 o 1 1 x = x + v t + at 2 θ = θ +ω t + αt 2 o o 2 o o 2 v2 = v2 + 2ax ω 2 = ω 2 + 2αθ o o 2nd Law of Motion ΣF F Στ τ a = = net α = = net m m I I Kinetic Energy and Power Translational kinetic energy Rotational kinetic energy 1 1 K = mv 2 K = Iω 2 2 2 Rolling object has both rotational and translational Copyright © 2013 National Math + Science Initiative®, Inc., Dallas, TX. All rights reserved. Rotation I: Rotational Kinematics & Energy kinetic energies 1 1 K = Iω 2 + mv2 2 2 Power Power FΔx τΔθ P = = Fv P = = τω t t Strategy on Rotation Problems 1. Draw a free-body diagram (FBD) for each object in the problem, showing the forces acting, the chosen coordinate system and the axes of rotation. 2. Write ΣF = ma for each translating object and Στ = Iα for each rotating object. 3. Relate the translational and rotational velocities and accelerations when applicable, like rolling without slipping or a string on a pulley without slipping. 4. If mechanical energy is conserved, then conservation of energy methods provide a useful method of calculating final linear and rotational velocities. Multiple Choice 1. A bowling ball of mass M and radius R rolls without slipping down an inclined plane as shown above. The inclined plane forms an angle θ with the horizontal. The coefficient of static friction between the ball and the plane is µs. Which of the following diagrams correctly shows the forces acting on the object? Copyright © 2013 National Math + Science Initiative®, Inc., Dallas, TX. All rights reserved. Rotation I: Rotational Kinematics & Energy Resolve forces on an c) correctly shows the forces. Weight always c incline - FBD pulls an object down, a normal force is always Direction of forces perpendicular to the surface, and friction is always parallel to the surface. In this case, friction points up the ramp because it opposes the motion of the bottom of the ball, causing the ball to roll. 2. An automobile moves in a circle of radius 110 meters with a constant speed of 33 m/s. What is the angular velocity of the car about the center of the circle in radians per second? a) 0.30 rad/s b) 0.30(π) rad/s c) 0.30(2 π) rad/s d) 110/33 rad/s v = 33m / s Given and Unknown Relationship between r = 110m a the linear and angular v = rω variables when an Equation v object is rotating ω = around a fixed axis. r 33ms / ω = Solution 100m ω = 0.30 rad/s Copyright © 2013 National Math + Science Initiative®, Inc., Dallas, TX. All rights reserved. Rotation I: Rotational Kinematics & Energy 3. Which of the following statements are true about an object’s rotational inertia? Select two answers. a. Rotational inertia is proportional to the object’s mass regardless of choice of axis. b. Rotational inertia is inversely proportional to the object’s speed. c. Rotational inertia has the units of kg.m/s2. d. Rotational inertia depends on the choice of the axis of rotation. Rotational Inertia for I = Σmr 2 a,d a solid (continuous Rotational inertia depends on the choice of axis mass distribution) of rotation, r. object Rotational inertia is proportional to the object’s mass regardless of choice of axis. 4. A wheel rotates through 10.0 radians in 2.5 seconds as it is brought to rest with a constant angular acceleration. What was the initial angular velocity of the wheel before the braking began? a) 0.25 rad/s b) 0.625 rad/s c) 2.0 rad/s d) 8.0 rad/s Angular kinematic 1 d θ = (ω + ω)t equations 2 o 2θ ω = − ω o t 2(10rad) ω = − 0 o 2.5s ω0 = 8rad / s 5. A solid disc has a rotational inertia that is equal to I = ½ MR2, where M is the disc’s mass and R is the disc’s radius. It is rolling along a horizontal surface with out slipping with a linear speed of v. How are the translational kinetic energy and the rotational kinetic energy of the disc related? a) Rotational kinetic energy is equal to the translational. b) Translational kinetic energy is larger than rotational. c) Rotational kinetic energy is larger than translational. d) The answer depends on the density of the disc. Copyright © 2013 National Math + Science Initiative®, Inc., Dallas, TX. All rights reserved. Rotation I: Rotational Kinematics & Energy Calculation of The ball will have both translational kinetic b mechanical energy energy and rotational kinetic energy. problem 1 Translational _ KE = Mv 2 2 2 1 2 1 ⎛ 1 2 ⎞⎛ v ⎞ Rotational _ KE I MR ⎜ ⎟ = ω = ⎜ ⎟⎜ 2 ⎟ 2 2 ⎝ 2 ⎠⎝ R ⎠ 1 Rotational _ KE = Mv 2 4 6. The discs shown above are connected by a belt that rotates both discs at the same time. The belt does not slip as it rotates the discs. The discs have different radii as shown, with R1 < R2. What is the relation between the angular velocities of the discs? a. ω1 = ω2 b. ω1R1 = ω2 R2 ω ω c. 1 = 2 R1 R2 R R d. 1 = 2 ω1 ω2 Relationships Both wheels have the same linear speed at their b between the linear outside edges because both wheels are being and angular variables rotated by the same belt that is moving without when an object is slipping. rotating around a fixed axis v1 = v2 ω1R1 = ω2 R2 Copyright © 2013 National Math + Science Initiative®, Inc., Dallas, TX. All rights reserved. Rotation I: Rotational Kinematics & Energy 7. A wrench has five equal forces V, W, X, Y, and Z applied to it. Which two forces produce the same clockwise torque? a) V and W b) V and X c) V and Z d) W and X τ = r F = rF sinθ ⊥ Definition of torque Both V and X produce a clockwise torque that is b equal in magnitude. V and Z produce a torque that is equal in magnitude, but Z produces a counterclockwise torque. 8. A square piece of plywood on a horizontal tabletop is subjected to the two horizontal forces shown above. Where should a third force of magnitude 5 newtons be applied so that the piece of plywood is in equilibrium? a. b. Copyright © 2013 National Math + Science Initiative®, Inc., Dallas, TX. All rights reserved. Rotation I: Rotational Kinematics & Energy c. d. The net force must equal zero, so the possible Torque and static answers are a and d. Answer d, however, would a equilibrium cause the plywood to rotate so the answer is a. 9. A door is free to rotate about its hinges. A force F applied normal to the door a distance x from the hinges produces an angular acceleration of α.
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