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Rotational (Angular) Motion Linear Motion vs. Rotational Motion Linear (Translational) Motion – involves an object moving in a straight line Rotation (Angular) Motion - involves an object rotating about an axis (examples: merry-go-round, rotating Earth, a spinning skater, a top, a turning wheel) Angular Quantities Where, 1 rad r = radius There is (m) 6.28 S =arc length (m) radians or 2π Θ = radian (unit less) radians in 1 circle s θ r 0.28 axis What is a radian? This is also A projection of 1 radius (or equivalent to 360° amount of angular displacement it takes to go) around a circle This equation relates linear and angular quantities! Arc Length (m) S = θr Angular Displacement (radians) Radius (m) Relate back to something you know! What do you call the arc length of the entire circle? θ 1 complete circle = 360° = 2 π radians r S = θr S = 2π r The circumference is basically the arc length of the entire circle C = 2π r The Radian! Is radian really a unit? NOPE! We only use radians to keep our “unit of measurement” in mind. (a.k.a make sure we are not in degrees) It is SUPPPER important you know how to change your calculator to and from radians - degrees Converting Quantities You need to know how to convert! 1 radian = ___ degrees? 180°= ___ radians? 4 revolutions = ___ radians? key concepts in rotational motion : its JUST like Linear motion! You can relate the two! Don’t let symbols intimidate you! Linear and Rotational Displacement (s and “x” or “d” have the same idea) Linear Rotational x x Linear and Rotational Displacement (s and “x” have the same idea) Linear S =θ r Arc Length (m) Up, N, E, right = + Down, S, W, left = - Rotational θ = S/r Angular Displacement (radians) CCW = + CW = - Linear and Rotational Velocity Radians per second (rad/s) Linear Rotational x x v= x = Δx = Δs t t t ω= = Δθ t Linear and Rotational Velocity Linear Rotational v = x = Δx = Δs t Δt Δt ω = θ = Δθ t Δt Measured in m/s Up, N, E, right = + Down, S, W, left = - v Measured in rad/s CCW = + CW = - ω Linear and Rotational Acceleration Linear Rotational Linear and Rotational Acceleration Linear a = Δv = vf -vi Δt Δt Measured in m/s2 Up, N, E, right = + Down, S, W, left = - Rotational α = Δω = ωf-ωi Δt Δt Measured in rad/s2 CCW = + CW = - Δs What do you know about the linear displacement? Angular? Δs Δs θ A B Objects C What do you know about the linear velocity? Angular? v = x = Δx = Δs t Δt Δt What do you know about the linear acceleration? Angular? a = Δv = vf -vi Δt Δt α ω = θ = Δθ t Δt = Δω = ωf-ωi Δt Δt There is a parallel between linear and rotational motion. Most of the equations will look “Familiar”, however, we will change linear values into angular values. The math is done the same way Rotational Kinematic Translations Every term in a given linear equation has a corresponding term in the rotational equation Angular Motion Graphs (resemble the motion graphs we saw during the kinematics unit) Angular Motion Graphs θ Δθ =ω t The SLOPE of a Angular Displacement vs Time graph is the Angular Velocity! Time Δω = α t ω Time The SLOPE of a Angular Velocity vs Time graph is the Angular acceleration! The Area under the line is the Angular Displacement! Area =ω t =Δθ Relationships between Linear and Angular Quantities Displacement S=θr Velocity v = ωr T Acceleration a = αr T Every point on a rotating object has the same angular motion Every point on a rotating object does not have the same linear motion For any circle you can convert to and from angular and linear velocity v = ωr ω = v/r We can apply the same principles to acceleration a = αr α = a/r Come in and get notebooks out… we have: NOTES! PROJECTS DUE TODAY BY 11:59 email to: [email protected] Rotational Dynamics In linear motion, and object requires a force to accelerate. In rotational motion, a torque is required Forces cause accelerations Torques cause angular accelerations Force and torque are related t Torque Torque, (greek letter “tau”), is the tendency of a force to rotate an object about some axis t=rF t = r F (sinθ) For Forces at angles τ r F Sin(θ) is torque is the length from the axis of rotation is the force is the angle between F and r Sign Convention for Torque Using the Right hand Rule!!! Positive torque: Counter-clockwise, out of page ccw cw Negative torque: clockwise, into page Things to know about Torque Torque is a Vector! It can be + or – Measured in Newton-meters (N.m) If the Force is at an angle, only the component PERPENDICULAR to the lever arm is applied. If more an one torque is applied, just add τ them as vectors. If net=0 , then there is no angular acceleration. Torque is determined by Three Factors: * The magnitude of the applied force. * The angle of the applied force. * The location of the applied force. The forces nearer the end of the wrench have greater torques. Location of force 20 N 20 N 20 N How we see Torque Torque – and the lever arm The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin q This also gives t = r F sin q τ net is Στ so…. Στ = τ1 + τ2 + ….etc. We can break it down further Στ = F1r1(sinθ) + F2r2(sinθ) +…etc 2 + - + 1 3 Στ = 30(2)(sin45°) + 25(0)(sin30°) - 10(2)(sin20°) A boy of mass 45 kg and his dog of mass 25 kg stand on a diving board. If the boy stands 2.0 meters from the pivot of the board, and the dog stands 1.2 meters from the pivot, what is the net torque on the board? Στ = τdog +τboy 2m τ = Fr 1.2m Στ = (-441*2)+(-245*1.2) 245 N 441 N Στ = -1176 N*m Come in.. Get both papers on the back of table… we got notes and a virtual lab today Rotational Equilibrium is a balance. When an object is in equilibrium the object is not accelerating. Two bubbles are on a seesaw. How far from the center does the yellow bubble have to sit to balance out the seesaw? Force of each. W = mg W = 294 N W = 196 N M= 30kg Set Torque equal on both ends. Fr 1 = Fr 2 M= (294)(3) = (196) (r2) 20kg Solve for r2 ?m 3m 882 = r 2 196 R2 = 4.5 meters What do we know so far? Angular Quantities S Θ r VT aT ω α τ Angular and Linear Formulas S=θr θ = s/r v = Δs Δt ω = Δθ Δt a =vf -vi Δt α = ωf-ωi Δt Angular and Linear Conversions v = ωr T Angular Kinematics a = αr T Torque t=rF t = r F (sinθ) Torque and Angular Acceleration When a rigid object is subject to a net torque (Στ ≠ 0), it undergoes an angular acceleration The angular acceleration is directly proportional to the net torque The more Torque, the more angular acceleration The relationship is similar to ∑ F = ma Newton’s Second Law Newton’s Second Law for a Rotating Object Net Torque Angular acceleration Moment of Inertia Moment of Inertia?? An objects resistance to change its rotation… similar to mass in linear motion. Moment of Inertia the moment of inertia depends on the quantity of matter and its distribution in the rigid object and its distance from the axis of rotation. Moment of Inertia Distance from axis kg.m2 Mass The units for Moment of Inertia is kg.m2 Moment of Inertia Objects with different rotational inertias move differently L m m 2 m1r m What is the moment of inertia for this? ΣΙ= ΣΙ= ΣΙ= 2 2 Σ Ι = ¼mL + ¼mL 2 Σ Ι = ½ mL 2 Σmr 2 2 + m2r + m3r 2 2 2 m(½L) + m0 + m(½L) L m m m ΣΙ= ΣΙ= 2 2 Σ Ι = ¼mL + mL 2 Σ Ι = 5/4 mL What is the moment of inertia for this? ΣΙ= 2 Σmr 2 2 2 m1r + m2r + m3r 2 2 2 m0 + m(½L) + m(L) Is the CENTER of MASS the same as the MOMENT of INERTIA? ΣΙ= 2 Σmr NO! X = m1x1 + m2x2 +…. (m1 + m2+….) CM The moment of inertia refers to the mass distribution of an object whereas the center of mass refers to the location where the mass is concentrated at. Ok, back to the 2nd law. Ex. Consider a solid cylinder of mass 3kg and radius of 2m. When a Force of 20 N is applied, what is the angular acceleration? F m r m = 3kg r= 2m F= 20 N I = ½ mr2 F r = ½ mr2 α - F = ½ mr α α = - 2F mr α = - 2(20) (3*2) α = -6.67 rad/s2 If the cylinder from the previous example started from rest… What is its angular displacement after 2 seconds? ωo = 0 rad/s t = 2s α = -6.67 rad/s2 θ=? θ = ωot + ½αt2 θ = (0*2) + ½ (-6.67 *22) θ = -13.34 rad τ What is the acceleration of the block? M (This problem uses both rotational and linear motion) r rotational m Linear M -T r m +T mg How to solve: 1. Draw FBD for all objects with mass. 2. Assign Direction 3. Relate α to α =aT /r 4. Combine equations and solve. mg - T= ma T r = ½ Mr2 α mg - ½ Ma = ma T = ½ Mr (a) r mg= ma + ½ Ma mg= a (m+ ½ M) T = ½ Ma mg (m+ ½ M) = a Try this one… What is the acceleration of m2 M1 r M M2 Rotational Kinetic Energy Rotating objects can have both rotational and translational kinetic energies Object will rotate through an angular displacement Both will result in kinetic energy Object will translate through a linear displacement KETOTAL = KE + KER KETOTAL = 2 ½mv “Rotational” + 2 ½Iω N FF What force causes Torque? FRICTION! FBD of an object rolling down a hill. The object will “slip” when going down the incline resulting in a change in angular velocity – and producing a torque. Does that mean rolling object move faster than a translating object? Object 1: Translating h No Friction Which will travel faster? Object 2: Rotating h Friction Both objects start out with the same potential energy =mgh. But object 1 will be faster because all the potential energy is converted to kinetic energy whereas in object 2 the total potential energy is split between kinetic energy and rotational kinetic energy. Object 1 Object 2 PE = KE PE = KE + KER mgh = ½mv2 mgh = ½mv2 + ½Iω2 v= 2𝑔ℎ v= 4 𝑔ℎ 3 Angular Momentum Linear Momentum Angular Momentum p = mv L = Iω τ = ΔL = Δmvr Δt Δt Angular Momentum is conserved unless a net external torque is present. A torque causes a change in momentum over time If Στ = 0, then Li = Lf If τ ≠ 0, then Li ≠ Lf