Rotational Motion

Rotational Motion Michelle Fitzsimmons Grade 12 Physics covs_mlf@access-k12.org November 6, 2006

Objectives • Compare and contrast motion in a straight line and rotational motion • Calculate angular displacement, velocity, and acceleration • Analyze how rotational motion can be transferred using gears, gear trains, and pulleys

Rectilinear Motion • Motion in a straight line • Studied previously • Newton’s Laws of Motion • Displacement, velocity, acceleration

Curvilinear Motion • Motion along a curved path • Example: satellite orbiting Earth

Rotational Motion Body itself is spinning

Measuring Circular Motion (Method 1) • 1 complete rotation = 1 revolution • Measure rotational motion by counting the number of rotations • Most common unit of measurement = revolution (abbreviated rev)

Measuring Circular Motion (Method 2) 360° • Can also measure using degrees • 1 rotation = 1 revolution = 360°

Measuring Circular Motion (Method 3) • Radian (rad): angle with vertex at center of circle whose sides cut off an arc on circle equal to its radius • Radius = r • Length of arc = r • angle = 1 radian • Unitless dimension – use rad to avoid confusion

Converting between different measurement • By definition: 1 rev = 360° = 2πrad • Convert 10πrad to revolutions 4 rev 4π rev 5 rev 5π rev

Sorry! TRY AGAIN!

Way to Go!

Angular Displacement • Distance through which any point on rotating body moves (angular distance instead of linear distance) • Example: when a wheel makes one complete rotation, it’s angular displacement has been 1 rev, 2πrad, or 360°

Angular Velocity • Similar to linear velocity, except instead of linear displacement, use angular displacement • Angular velocity = rev/time • OR ω = angular displacement (θ)/time (t) • Units = rad/s or rev/min θ t ω =

Example • A motorcycle wheel turns 7200 times while being ridden for 10 min. What is the angular velocity in rev/min? 720 rev/min (rpm) 700 rev/min (rpm) 500 rev/min (rpm) 600 rev/min (rpm)

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Linear Velocity • The linear speed on any point on rotating circle = v = ωr v = linear velocity ω = angular velocity r = radius • Can measure using stroboscope or strobe light

Example: Linear Velocity A wheel of 1.00 m radius turns at 1000 rpm. Find the linear speed of a point on the rim of the wheel. 6000 m/s 600 m/s 53 m/s 105 m/s

Way to Go!

Angular Acceleration Angular acceleration = rate of change of angular velocity α = angular acceleration Δω = change in angular velocity t = time Δω t α =

Equations: Linear vs. Rotational Motion

Sample Calculations • To calculate various components of angular motion, click on the link listed below: http://canario.iqm.unicamp.br/MATDID/HyperPhysics/hbase/rotq.html

Transferring Rotational Motion • Rotational motion can be transferred using gears or belt-driven pulleys • This is how motor-driven vehicles operate • Use to reduce or increase angular velocity of rotating shaft or wheel • Larger diameter = slower rpm (rev/min)

Finding new rpm: Gears T∙N = t∙n T = # teeth (driver gear) N = rpm (driver gear) t = # teeth (driven gear) n = rpm (driven gear)

Finding new rpm: Gear Trains • Gear Trains • If total number of gears is odd – the first and last gear turn in the same direction • If total number of gears is even – the first and last gears turn in opposite directions • NT1T2T3… = nt1t2t3… • N = rpm of first driver gear • T = # of teeth in driver gears • N = rpm of last driven gear • T = # of teeth in driven gears

Finding new rpm: Pulleys D∙N = d∙n D = diameter (driver) d = diameter (driven) N = rpm (driver) n = rpm (driven) For series of pulleys: ND1D2D3…= nd1d2d3…

For more information, click on the following links: • http://canario.iqm.unicamp.br/MATDID/HyperPhysics/hbase/rotq.html • http://www.howstuffworks.com/gear.htm • http://www.physicsclassroom.com/mmedia/circmot/circmotTOC.html • http://theory.uwinnipeg.ca/mod_tech/node42.html

Credits (in order of appearance) • http://www.ntsg.umt.edu/graphics/ • http://www.blackforestgifts.com/detail.aspx?ID=4045 • http://www.mathsisfun.com/geometry/radians.html • http://www.seykota.com/tribe/pages/2003_Dec/Dec_21-30/index.htm • http://lancet.mit.edu/motors/motors3.html • http://www.howstuffworks.com/gear.htm • http://www.dodge-pt.com/products/gearing/pulleys/pulleys.html • http://www.deltacad.com/sample/gears.html • http://engines.rustyiron.com/

Angular Momentum • Moment of inertia • Rotational inertia • Property of a rotating body that causes it to continue to turn until a torque causes it to change its rotational motion • Torque is produced when a force is applied to produce rotation (τ = Fst)

Newton’s 2nd Law (Rotational Equivalent) • Angular acceleration is directly proportional to torque τ = Iα τ = applied torque I = moment of inertia α = angular acceleration • Applies to rigid body rotating about a fixed axis

Moment of Inertia • Inertia: depends on mass and how far mass is from center of rotation • Hollow = more inertia, mass concentrated further from center (both have same mass)

Momentum • Linear momentum: measure of amount of inertia (p = mv) • Angular Momentum L = IωL= angular momentum I = moment of inertia ω = angular velocity

Angular Impulse • Change in angular momentum τt = Iωf – Iωi τ= torque t = time I = moment of inertia ωf = final angular velocity ωi = initial angular velocity

Linear/rotational comparison

Conservation of Angular Momentum • The angular momentum of a system remains unchanged unless an external torque acts on it • Spinning ice skater • Arms extended, inertia is larger, velocity is smaller • Pull arms in tight, inertia decreases, velocity increases

Centripetal Force • Centripetal force acting on a body in circular motion causes it to move in a circular path • Force is exerted toward center of circle F = mv2F = centripetal force r m = mass v = velocity r = radius of curvature

Power in Rotational Systems • Used when engine or motor turns a shaft • Earlier, we learned P = Fs/t = W/t (linear) • Rotational power P = (torque)(angular displacement) time = torque x angular velocity = τω

Rotational Motion