note: 2 radians = 1 rotation or revolution = 360 degrees. …...rotational ke ke rot = ½ i ’2...
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Note:2𝜋 radians=1rotationorrevolution=360degrees.Theseareconversionfactors.
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Fidgetspinnerexample
𝑠 = 𝜃𝑟
Angularspeed(𝜔 ) ofafidgetspinnerspinningat62.5Hz– converttoradianspersecond
62.5𝑟𝑜𝑡𝑠∗ 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛
= 393𝑟𝑎𝑑/𝑠𝑒𝑐
If1rotationofthefidgetspinner=2𝜋 radians,angulardisplacementofthespinnerasonearmmovestothepositionofthe2nd arm:𝜃 = 1/3(2𝜋)radians
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If centripetal force keeps an object in rotation, what force causes rotation or
slows rotation down?
Torque = “Twisting force”Torque = force times lever arm (or radius)
𝝉= r ´ F
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Why hobbit doors are a stupid designWhereyouapplyaforcetocauseatorquemattersApplyaforceclosertothehinge,thedoordoesn’topenasquickly– smallerr(leverarm),sosmallertorque.
Onlytheperpendicularcomponentoftheforcewillcontributetorotation𝝉= rFsin𝜽𝜃 (in degrees) is the angle between force applied and lever arm.
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The Hulk and Hercules are in competition for some reason. They apply forces to 2 thin cylindrical wheels of radii r1 = 3.0m and r2 = 5.0 m that are attached to each other on an axel that passes through the center of each as shown. Who wins and what’s the net torque?
𝝉hulk = (5.0 x 106 N cos30)(5m) = 2.2 x 107 Nm𝝉herc = (5.0 x 106 N)(3m) = 1.5 x 107 NmNet 𝝉 = 2.2 x 107 Nm - 1.5 x 107 Nm
Net 𝝉 = 6.7 x 106 Nm clockwise : Hulk wins
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Just like how Force causes linear acceleration, Torque causes angular acceleration
𝛴𝐹 = 𝑚𝑎𝛴𝜏 = rma𝛴𝜏 = rmr𝛼
𝛴𝜏 = (𝛴𝑚𝑟D)𝛼𝛴𝜏 = 𝐼𝛼
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Rotational Inertia (or “Moment of Inertia”) (I)
A measure of an object’s “laziness” to changes in rotational motionGeneral equation :
I = 𝛴mr2
Depends on mass ANDdistance of mass from axis of rotation
The farther away the mass is from the center, the more difficult it is to rotate an object
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Rotational Inertia (or “Moment of Inertia”) (I)
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QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Balancing Pole increases Rotational Inertia
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Angular MomentumMomentum resulting from an object moving in linear motion is called linear momentum. Momentum resulting from the rotation (or spin) of an object is called angular momentum.
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Just like how Force causes a change in momentum,
𝐹𝑡 = 𝑚Δ𝑣 →F =Δp/Δt (Newton’s 2nd Law: Momentum format)
𝝉 causes 𝜟L (change in angular momentum)𝜏 = ΔL /Δt
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Calculating angular momentum
Angular momentum is calculated in a similar way to linear momentum, except the mass and velocity are replaced by the moment of inertia and angular velocity.
Angularvelocity(rad/sec)
Angularmomentum(kg m/sec2)
L = I w
Moment of inertia(kg m2)
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Conservation of Angular Momentum
Angular momentum is important because it obeys a conservation law, as does linear momentum. If there is no net torque acting on a system, angular momentum is conserved.
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GyroscopesFidget spinner can act like a gyroscope : Spinning fidget spinner is easier to balance on a pencil point than a spinner at rest.
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Rotational KEKErot =½I𝜔2
TotalKE=KEtrans +KErotForanyrollingobjectofmassmandradiusr,whatisitsspeedatthebottomofthehill?PEi =Ketrans, final +KErot,finalmgh =½mv2 +½I𝜔2 Iforanyrollingobject=cmr2 𝜔 = K
Lmgh =½mv2 +½(cmr2)(v2/r2)gh =½v2+2/10v2
v= DMNOPQ
�greaterrotationalinertia=smallerspeed.Speeddoesn’t
dependonmassorradius.
Cissomeconstant
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Rotational KEWillacylinderofsouporacylinderfilledwithamoresolidmaterialreachthebottomfirst?
liquid:theinsidesslidemorethantheyrotate,solessoftheinitialpotentialenergygoesintorotationalkineticenergy,andmorecangointotranslational,movingthecandowntheincline.