Rotational Motion Chapter 7
Rotational Motion • Motion about an axis of rotation. • A record turntable rotates; • A bug sitting on the record revolves around the axis and is said to undergo circular motion.
Spin cycle of washer • The spin cycle of your washer works on the principle that your clothing is forced to follow a circular path, but the water in the clothing escapes through holes in the side of the drum, not following a circular path.
Measuring rotational motion: • You have probably already encountered the radian, the measure of angular displacement: • Angle whose arc length = its radius • ΔΘ = Δs/r • Θ is anglular in radians, s is arc length, r is radius • Converting to degrees: • 2 π(rad) = 360 (deg)
Just like linear displacement, the direction matters. • Conventionally, rotation is….. • Positive when Counterclockwise • Negative when Clockwise • Lets do the problem on P246…. • H/W P247 Q1-4
P247 answers • 1. 1.7 rad • 2. Pi rad, 1.2 m • 3. 0.34 rad • 4. 2.5 rad, 6.4m, - 320 ° , 1.1m
Angular velocity: • Think of it as how quickly something is turning • A unit that is often used is revolutions per minute (RPM) • Old records spun at 33.3, 45 or 76 RPM • Car engines often run most efficiently at about 2500 RPM and produce the maximum power about 4500rpm • Most electric motors spin at a multiple or sub multiple of 3600RPM or 60 revolutions/sec
Angular velocity (speed) • Just like motion in a straight line, after displacement comes speed …. • Angular speed is the rate of change of angular displacement • ω = ΔΘ/Δt • Units are rad/s • Lets do problem on P248 • H/W P248 Q1-4
P248 answers 1. 29 rad/s • 2.2 rad/s • 7.3 X 10-5rad/s • A) 0.23 rad/s b) 0.24 rad c) -6.3 rad/s d) 0.75s
Angular acceleration: • Think of it as how quickly a rotating object speeds up or slows down. • The angular acceleration of the earth is High:Low: Zero • The angular acceleration of a bicycle wheel is pulling away from a stop High:Low:Zero • The angular acceleration of a motorcycle doing a constant 150mphis High:Low:Zero • The angular acceleration of a motorbike wheel pulling away from a race start is High:Low: Zero
Angular acceleration • Rate of change of angular velocity, α α = ωf – ωi Δt • Units are rad/s2 • Let’s do Problem on P249 • H/W P. 250 Q1,2,3
P250 Answers • 4.3rad/s2 • 1.3rad/s2 • a) 17rad/s2 b) 0.038rad/s c) -6.3 rad/s2
Angular kinematic equations: ωf = ωi + α tθ = ωit + 1/2 α t2ωf2 = ωi2 + 2 α (θf-θi) θ =1/2 (ωi + ωf) t
Answers to P 252 • 9.0 rad/s • 25 rad/s2 • 15 rad/s • 31 rad/s • 0.89 rad/s
Section review: • 1. 0.44rad, 0.61 rad, 2.23 rad, 4.7 rad • 2. -1.0 rad • 3. 0.314 rad/s • 4. 0.20 rad/s2 • 5. 0.70 rad/s • Page 269 Q10: 0.042rad/s , Q11a) 821rad/s2 , b) 4.2 X103 rad
Remember the strategy: • Write down the givens and unknown. • Find the equation that has all the givens and unknown and nothing else. • If necessary, rearrange the equation to find the unknown • and then substitute to solve.
Tangential Speed (7.2) • Speed of an object (m/s) traveling in a circle is called Tangential Speed because the direction of motion is always in a tangent to the circle.
Tangential speed: Tangential speed would be important to find out how fast a point on the earth is travelling in a given time etc vt (m/s)= r ω
Tangential Acceleration: • The rate of change of tangential speed. It is the linear acceleration of a point undergoing angular acceleration: at (m/s2) = r α
Centripetal acceleration: • Acceleration directed toward the center of a circle that an object undergoing circular motion must experience. (Note spinning cup with water in it) ac= vt2 / r ac= r ω2
H/W : P255 1-4, P256 1-3, P258 1-5 • P250 • 1.8m/s • 6.9 m/s • 9.2 m/s • 3.6 m/s, 15 rad/s, 29m/s, 1.3m • P256: • 2.11 m/s2, 0.18m/s2, 1.0m/s2
P258 answers: • 3.0m/s2 • 250m/s2 • 1.5m/s, 1.0rad/s • 12.6m/s2 • 84m/s2
Centripetal force: • In order for an object to travel in a circle, something must provide a force that is directed at all times toward the center of the circle. This force is called CENTRIPETAL FORCE. • For a car going around the corner, the force is provided by the ______. • For a stone being twirled in a slingshot it is provided by the _______. • For clothes in the spin cycle it is provided by______
For the moon traveling around the earth it is provided by _______ . • For the earth traveling around the sun it is provided by _______ . • Can you think of any other objects that undergo circular motion and identify what provides the centripetal force?
Demonstration • The object on the left travels with inertia, while the object on the right is caused to travel in a circle by the wooden block. Centripetal force is applied.
Inertia should cause the car to continue in the direction in which it was traveling. What causes it to travel in a circular direction? What applies the centripetal force?
If you let go, you’ll be like Mary Poppins and fly off the Merry-go-Round.
You do not fly straight outward. • Instead you follow tangential motion, and continue in a straight line from the point where the circular motion ends.
As usual, there is a formula: (From F=ma) Fc= mvt2 / r Fc= mrω2 Homework: P261 Q1-5
Newton’s universal law of gravitation • “There is an attractive force between any two masses or particles in the universe” F = - G m1m2 r2 Where G is the universal gravitational constant, m is each mass in kg, and r is the distance separating their centers of mass G = 6.67 X 10 -11 N m2 / kg2
P265 1-3 top of page • And Section review • Keep in mind that, for an orbiting body, centripetal force = gravitational force.
Speed of an orbiting satellite: • Vs = (G Mc /r)1/2 • Where Mcis central mass, r is the total distance from center of rotation.
Escape Velocity: • There is a speed at which an object shot straight up from a planet will have enough energy to escape the gravitational field of the planet. • Vesc = ( 2MG/R)1/2 • M is the mass of the planet.
g, the acceleration due to gravity on any planet surface : • g = G Mp/ rp2
Homework: • Find your gravitational force on the earth’s surface using universal G formula. (1lb = 0.45kg) • Compare with the weight formula result. • Compare with your gravitational force in orbit 300km above the earth’s surface. • Find g for each planet and the moon. • Find the escape velocity for each planet and the moon.
Rotational Speed (angular speed) • The number of rotations/unit of time. • RPM = rotations/min
Centripetal Force • Centripetal force is a force that causes an object to travel in a circle.
Centrifugal Force • Centrifugal means “Center-fleeing” and it is a force that seems to push you outward. • Think playground “Merry-go-Round”
What it really is is inertia. • Newton’s First Law applies always.
Inertia, Centrifugal Force • In a car.
Kids, Don’t try this at home… • Experts state that you can swing a bucket of water over your head and it won’t fall out because of centrifugal force (INERTIA). • What they don’t say is that when you stop swinging, it will drench you!
The breaking string revisited… • What kind of tension would be in that string?
Rotational Mechanics • Torque – Rotational analog of Force; • Produces rotation • More “leverage” = More Torque
What is Torque?? • Used when you use a hammer claw to remove a nail • Used when you use a long-handled wrench to loosen a bolt • The longer the handle, the greater the torque
Important facts to increase Torque • The force must be applied perpendicular to the plane. • The Longer the Lever, the greater the force.