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Rotational Motion

Rotational Motion. Chapter 7. Angles. Been working with degrees for our angles 90 degrees, 180, 56.4, etc. There is another way to measure an angle, which is called radians. Radians. Radians are found by the following: Θ =(s/r) s is the arc length of the circle

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Rotational Motion

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  1. Rotational Motion Chapter 7

  2. Angles • Been working with degrees for our angles • 90 degrees, 180, 56.4, etc. • There is another way to measure an angle, which is called radians

  3. Radians • Radians are found by the following: Θ=(s/r) • s is the arc length of the circle • r is the radius of the circle • Radians are usually some multiple of pi.

  4. Unit circle

  5. Radians vs. degrees • 360 degrees is the same as 2π radians -Degree to radian: radian = (π/180) * degree -Radian to degree: degree = (180/π) * radian One revolution = 2π radians = 360 degrees Convert: 35 degrees to radians 5.6π radians to degrees

  6. Angular displacement • Angular displacement is how much an object rotates around a fixed axis • Such examples would be a tire rotating, or a Ferris wheel car.

  7. Angular displacement • Finding angular displacement is simply a matter of finding the angle in radians: Δθ=(Δs/r) • So the change in angular displacement is equal to the change in arc length over the radius.

  8. Sample Problem • A Ferris wheel car travels an arc length of 30 meters. If the wheel has a diameter of 45 meters, what is the car’s displacement?

  9. Angular speed • Angular speed is how long it takes to travel a certain angular distance. • Similar to linear speed, angular is found by: ωavg= Δθ/Δt and its units are rad/s, though rev/s are often used as well

  10. Sample Problem • An RC car makes a turn of 1.68 radians in 3.4 seconds. What is its angular speed?

  11. Angular acceleration • Lastly, angular acceleration is how much angular speed changes in that time interval. αavg=(ω2-ω1)/Δt The units are rad/s2 orrev/s2, depending on angular velocity

  12. Sample problem • The tire on a ‘76 Thunderbird accelerates from 34.5 rad/s to 43 rad/s in 4.2 seconds. What is the angular acceleration?

  13. Episode V: Kinematics Strike Back • Displacement, speed, acceleration…should all sound familiar • Recall the linear kinematics we discussed earlier.

  14. Linear vs. Angular • Linear and angular kinematics, at least in form, are very similar.

  15. NOTE • These kinematic equations only apply if ACCELERATION IS CONSTANT. • Additionally, angular kinematics only for objects going around a FIXED AXIS.

  16. Sample problem • The wheel on a bicycle rotates with a constant angular acceleration of 3.5 rad/s2. If the initial angular speed of the wheel is 2 rad/s, what’s the angular displacement of the wheel in 2 seconds?

  17. Tangential & Centripetal Motion • Almost all motion is a mixture of linear and angular kinematics. • Reflect on when we talked about golf swings in terms of momentum and impulse.

  18. Tangents • A tangent line is a straight line that just barely touches the circle at a given point.

  19. Tangential Motion • Similarly, for an instantaneous moment in circular motion, objects have a tangential speed. • So for an infinitesimally small time, an object is moving straight along a circular path.

  20. Tangential speed • Tangential speed depends on how far away the object is from the fixed axis.

  21. Tangential speed • The further from the axis you are, the slower you will go. • The closer to the axis you are, the faster you will go.

  22. Tangential speed • So, during a particular (infinitesimally small) time on the circular path, the object is moving tangent to the path. • No circular path, no tangential speed

  23. Tangential speed • The tangential speed of an object is given as: vt=rω where r is the distance from the axis, or the radius of a circle. Remember, the units for linear speed is m/s.

  24. Sample problem If the radius of a CD in a computer is .06 m and the disc turns at an angular speed of 31.4 rad/s, what’s the tangential speed at a given point on the rim?

  25. Tangential acceleration Of course, where there is speed, there probably is also acceleration But keep in mind: THIS IS NOT AN AVERAGE ACCELERATION.

  26. INSTANTANEOUS Tangential Acceleration • Tangential acceleration also points tangent to the circular path, found by: at=rα

  27. Sample Problem • What is the tangential acceleration of a child on a merry-go-round who sits 5 meters from the center with an angular acceleration of 0.46 rad/s2?

  28. Centripetal Acceleration • You can make a turn at a constant speed and still have a changing acceleration. Why?

  29. Centripetal Acceleration • Remember, acceleration is a VECTOR, just like velocity. • So when you’re pointing in a different direction along a circular path, acceleration is changing, even though velocity is constant. • This is known as centripetal acceleration.

  30. Centripetal Acceleration • Centripetal acceleration points TOWARDS the center of the circular path.

  31. Centripetal acceleration • There are two ways to determine this acceleration: ac=vt2/r OR ac=rω2

  32. Sample problem A race car has a constant linear speed of 20 m/s around the track. If the distance from the car to the center of the track is 50 m, what’s the centripetal acceleration of the car?

  33. Acceleration • Centripetal and tangential acceleration are NOT IDENTICAL. • Tangential changes with the velocity’s magnitude. • Centripetal changes with the velocity’s direction.

  34. Total Acceleration • Finding the total acceleration of an object requires a little geometry.

  35. Causes of circular motion

  36. Circular Motion • If you’ve ever gone round a sharp turn really fast, you probably feel yourself being tilted to one side. • This is due to Newton’s Laws

  37. Back to THOSE… • Objects resist changes in motion. • When you go round a curve, your body wants to keep going in a linear path but the car does not.

  38. Once more… • So for a linear path, if F=ma, then for a circular path, Fc=mac • This is known as centripetal force.

  39. Centripetal Force • There are two other ways to find this force. Fc=(mvt2)/r OR Fc=mrω2

  40. Sample problem A 70.5 kg pilot is flying a small plane at 30 m/s in a circular path with a radius of 100 m. Find the centripetal force that maintains the circular motion of the pilot.

  41. Conundrum • Centripetal force points towards the center of the axis. • BUT in a car, you feel like you’re being flung AWAY from the center of axis. • So, what gives?

  42. When in doubt, Newton • Your body’s inertia wants to keep going in a linear direction. Which is why you tend to tilt away from the center of axis on a curve. • This is often labeled as centrifugal force, but it is NOT a proper force.

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