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Kinematics vs. Rotational Motion. When discussing kinematics, what are some common vector quantities we discuss? Displacement Velocity Acceleration Force Momentum
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Kinematics vs. Rotational Motion When discussing kinematics, what are some common vector quantities we discuss? • Displacement • Velocity • Acceleration • Force • Momentum ALL of these vectors can be described in terms of rotational motion! Angular displacement Angular velocity Angular acceleration Torque Angular momentum
How do we describe angular displacement? Lets say I have a disk rotating clockwise… Ex: The dot has an angular displacement of 90o or π/2 rad or 1.57 rad • From math class, how do we describe the amount a circle rotates? • Radians (rad) • Symbol: θ r
What exactly is 1 radian? • One radian is the angle created by an arc whose length (displacement) is equal to the radius. • When x = r the angular displacement is oneradian. Therefore,
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. • How many degrees is this? • How small an object can the bird distinguish when flying a height of 100 m? a. b.
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. Symbol: ω (omega) Units: rads/sec
How is angular velocity related to linear velocity? Can we express both in one equation? • Yes, we can combine equations and simplify using our expression for radians! • Related:
Angular Acceleration How did we determine angular velocity? We simply applied the linear velocity equation in terms of radians! So what is angular acceleration? Symbol: α (alpha) Units: rad/sec2 Look familiar?
Tying together relationships What expression related angular displacement to displacement and angular velocity to velocity? How is each relationship similar? So, what do you think is the equation that relates angular and linear acceleration?
How do all the kinematic equations relate to rotational motion? Kinematics Rotational
Making things even easier! A lot of the time, rotational questions give you the angular velocity as “rpm” or “rps”. We can convert this since one rotation is 2π radians! What is 5.5 rpsin rad/s? • “rps” is simply frequency, so this conversion can be written as:
Example A centrifuge rotor is accelerated from rest to 20,000 rpm in 30 s. • What is the average angular acceleration? • Through how many revolutions has the centrifuge rotor turned during its acceleration period, assuming constant angular acceleration?
Solutions • ωi = 0, so that leaves just ωf ωf=2πf = 2π rad/rev x 333.3 rev/s ωf ≈ 2100 rad/s α = 70 rad/sec2 b. Recall one revolution is 2π radians. θ = 0 + .5(70 rad/s2)(30s)2 θ = 3.15 x 104 rad 5000 rev
What is Newton’s Second Law? • F = ma So, what did we say is the rotational equivalent to force? • TORQUE!!! We learned about torque earlier, but how can we describe it in terms of rotational motion? How did we write the rotational versions for ALL the kinematic equations??? This is the exact same equation that we learned before, but now we see how it matches all the other rotational ones!
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 • F = m x rα • τ = mr2α Now we can substitute into the equation for torque!
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”
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 halfway between the weights. I = Σmr2 I = (5kg)(2m)2 + (7kg)(2m)2 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. I = Σmr2 I = (5kg)(.5m)2 + (7kg)(4.5m)2 I = 143 kg.m2 How does the inertia added by the mass close to the axis compare to the mass farther away?
How is angular momentum represented? In order to answer this question we need to ask ourselves how linear momentum is represented. Rotational Momentum L = Iω ΔL = Στ x Δt Linear Momentum p = mv J = F x t = Δp Symbol: L Units: Whatever works
Conservation of Angular Momentum Without an acceleration can there be a force? Without an angular acceleration can there be a torque? If there is no torque on an object, will the angular momentum change? • Similar to the linear case, without a change in torque or angular velocity there can not be a change in “angular momentum”. It remains constant! • Ioωo = Ifωf = constant
Angular Momentum Demo • How is angular momentum conserved? • What factors are changing to conserve momentum? • Once spinning is any additional torque or angular acceleration provided?
Practice Angular Momentum Astronauts use a centrifuge to simulate the harshness of a shuttle launch. Initially the centrifuge has a length of 8m and an angular velocity of 10 rad/sec. The machine is then pulled in to half it’s length while spinning. What is the new angular velocity? Ioωo = Ifωf mr2ω = (mr2)fωf 82(10) = 42ωf 640 = 16ωf ωf = 40 rad/sec
Rotational Kinetic Energy and the Moment of Inertia Rotational KE can be written as follows Where I, the moment of inertia, is given by
Conservation of Energy Recall: The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:
Example 1 Solve for the final velocity of the bowling ball if the its moment of inertia is given by
Does the disk reach the bottom of the plane (a) before (b) after, or (c) at the same time as the hoop? Ihoop = MR2 Idisk = ½ MR2
Who wins? (a) before Both objects have the same potential energy, U = MGH. More of the potential energy goes into rotational kinetic energy for the hoop, because it has a larger moment of inertia. That leaves less translational energy for the hoop, so it moves slower.
