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Kinematics vs. Rotational Motion When discussing kinematics, what are some common vector quantities we discuss? • Displacement Angular displacement • Velocity Angular velocity • Acceleration Angular acceleration • Force Torque • Momentum Angular momentum ALL of these vectors can be described in terms of rotational motion! How do we describe angular displacement? Lets say I have a disk rotating clockwise… From math class, how do we describe the amount a circle rotates? • Radians (rad) • Symbol: θ r Ex: The dot has an angular displacement of 90o or π/2 rad or 1.57 rad What exactly is 1 radian? • One radian is the angle created by an arc whose length is equal to the radius. When l = r the angular displacement is one radian. Therefore, l q= r Make the following conversions 1 revolution to radians 60 degrees to radians 4.5 revolutions to radians 48 degrees to radians 2π rad 1.05 rad or π/3 rad 9π rad or 28.3 rad 0.84 rad A falcon can distinguish objects that extend a minimum angular displacement of 3 x 10-4 rad. a. How many degrees is this? b. How small an object can the bird distinguish when flying a height of 100 m? a. o 360 (3´10-4 rad)´ = 2p rad b. l =100m´ (3´10-4 rad) l = 3´10-2 m 0.017o How do we describe “angular velocity”? Thinking about velocity for a second, how do we determine velocity? • Displacement/time = velocity So it should be intuitive that angular velocity is how many radians are covered in a certain period of time. q Symbol: ω (omega) w= Units: rads/sec t How is angular velocity related to linear velocity? w= q t l v= t Can we express both in one equation? • Yes, we can combine equations and simplify using our expression for radians! • Related: v = rw l q= r l = rq Angular Acceleration How did we determine angular velocity? We simply applied the linear velocity equation in terms of radians! So what is angular acceleration? w - wi D w f Symbol: α (alpha) a = = t t Units: rad/sec2 Look familiar? Tying together relationships What expression related angular displacement to displacement and angular velocity to velocity? l q = ® l = rq r v = rw So, what do you think is the equation that relates angular and linear acceleration? How is each relationship similar? a = ra How do all the kinematic equations relate to rotational motion? Kinematics Rotational q d v t v 2f vi2 2ad t 2 2 w f = wi + 2aq v f vi at w f = wi + a t v vi v f 2 1 2 d vi t at 2 w= w= wi + w f 2 1 2 q = wi t + a t 2 Making things even easier! rev 2p rad rad A lot of the time, rotational 5.5 ´ = 34.56 sec 1rev sec questions give you the angular velocity as “rpm” or • “rps” is simply frequency, “rps”. so this conversion can be written as: w = 2p f We can convert this since one rotation is 2π radians! What is 5.5 rps in rad/s? Example A centrifuge rotor is accelerated from rest to 20,000 rpm in 30 s. a. What is the average angular acceleration? a. Through how many revolutions has the centrifuge rotor turned during its acceleration period, assuming constant angular acceleration? Solutions a. ωi = 0, so that leaves just ωf rev 1min rev 20, 000 ´ = 333.3 min 60sec sec ωf=2πf = 2π rad/rev x 333.3 rev/s ωf ≈ 2100 rad/s w f = wi + a t α = 70 rad/sec2 b. Recall one revolution is 2π radians. 1 2 q = wi t + a t 2 θ = 0 + .5(70 rad/s2)(30s)2 θ = 3.15 x 104 rad 1rev 3.15´10 rad ´ = 5000 rev 2p rad 4 Examples Summary 1. How is linear acceleration, velocity and displacement related to it’s rotational counterpart? 2. How is frequency related to angular velocity? 3. What is “rps” and “rpm” stand for? What do both represent? How do we describe Newton’s second law for rotation? What is Newton’s Second Law? • F = ma So, what did we say is the rotational equivalent to force? • TORQUE!!! Weislearned about torque earlier, but This the exact same equation that wehow can we describe it in terms of rotational motion? How did learned before, but now we see how it we writeallthe versionsones! for ALL the matches therotational other rotational kinematic equations??? t = rF Deriving a rotational expression for torque So now we have “τ = rF”, but if torque represents the 2nd law for rotation, then what about mass? How can we write force in terms of rotation? • F = ma Now we can substitute into • F = m x rα the equation for torque! • τ = mr2α Rotational Inertia τ = mr2 x α So, lets compare this to F = ma • Torque is rotational force • Angular acceleration is rotational acceleration • (mr2) is rotational inertia with symbol “I” I = å mr 2 t = rF ® t = mr a ® å t = I a 2 Demo: Rotational Inertia Which has more rotational inertia “I”? • Rotational motion measures how hard it is to change angular velocity. • It’s based on mass and it’s distribution regarding the axis of rotation. The cylinder is faster, so it must have less rotational inertia. It was easier to move! Two weights on a bar: different axis, different “I” Two weights of mass 5 kg and 7 kg are mounted 4 m apart on a light rod (whose mass can be ignored). Calculate the moment of inertia when rotated about an axis 2 I = Σmr halfway between the I = (5kg)(2m)2 + (7kg)(2m)2 weights. I = 48.0 kg.m2 Different Axis Calculate the moment of inertia now when rotated about an axis 0.5 m to the left of the 5 kg mass. How does the inertia I = Σmr2 I = (5kg)(.5m)2 + (7kg)(4.5m)2 added by the mass close to the axis . 2 I = 143 kg m compare to the mass farther away? Summary 1. What is Newton’s second law for rotational motion? 2. How can we define rotational inertia? 3. What affects the rotational inertia “I”? 4. What was the difference between parts “a” and “b” in the practice questions?