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Measuring Rotational Motion

Measuring Rotational Motion. Chapter 7 Section 1. What is Rotational Motion?. Rotational Motion – The motion of a body that spins about an axis. Examples: Ferris Wheel Bicycle Wheel Merry-go-round Etc…. Circular Motion.

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Measuring Rotational Motion

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  1. Measuring Rotational Motion Chapter 7 Section 1

  2. What is Rotational Motion? • Rotational Motion – The motion of a body that spins about an axis. • Examples: • Ferris Wheel • Bicycle Wheel • Merry-go-round • Etc…

  3. Circular Motion • A point that rotates about a single axis undergoes circular motion around that axis. • Regardless of the shape of the object, any single point on the object travels in a circle around the axis of rotation.

  4. Linear motion here? Think again • Its difficult to describe the motion of a point in a circular path through linear quantities, because the direction of the motion is not in a straight line path. • Therefore it is describe in terms of angle through which the point on an object moves.

  5. Rotational Motion • When rotational motion is described using angles, all points on a ridged rotating object, except the points on the axis move through the same angle during any time interval.

  6. r s θ Reference Line • r = radius • θ = angle • s = arc length

  7. Radians • Radian – An angle whose arc length is equal to its radius, which is approximately equal to 57.3º • The radian is a pure number with no dimensions.

  8. Radian Chart

  9. Radian Equations • Any angle θ, measured in radians, is defined as the following: • To convert degrees to radians, or vise versa, use the following equation:

  10. Angular Displacement • Angular Displacement – The angle through which a point, line, or body is rotated in a specific direction and about a specified axis. • In simple terms, angular displacement describes how much an object has rotated. • It is not a distance!

  11. Angular Displacement Equation • Δθ = Angular Displacement (rad) • Δs = Change in Arc Length (m) • r = Distance from Axis (m) • When Δθ rotates: • Clockwise – Negative • Counter-clockwise – Positive

  12. Recall • Recall that Δ is a capital Greek letter, “delta” • It means, “Change in” • It is always the final minus the initial • Example: • Δθ = θf-θi

  13. Example Problem • Earth has an equatorial radius of approximately 6380km and rotates 360º in 24 hours. • What is the angular displacement (in degrees) of a person standing on the equator for 1.0 hour? • Convert this angular displacement to radians • What is the arc length traveled by this person?

  14. Example Problem Answer • 15º • 0.26 rad • Approximately 1700km

  15. Example Problem • Earth has an average distance from the sun of approximately 1.5x10^8 km. For its orbital motion around the sun, find the following: • Average daily angular displacement • Average daily distance traveled

  16. Example Problem Answer • 1.72x10^-2 rad • 2.58x10^6 km

  17. Angular Speed • Angular Speed – The rate at which a body rotates about an axis, usually expressed in radians per second. (rad/s) • Recall that linear speed describes the distance traveled in a specific time, here it is the amount of rotation in a specific time. • It describes how quickly the rotation occurs.

  18. Angular Velocity Equation • ω= Angular Velocity (rad/s) • Lower case Greek letter called, “omega” • Δθ= Angular Displacement (rad) • t = Time Interval (s)

  19. Revolutions • Occasionally angular speeds are given in revolutions per unit time. • 1 rev = 2πrad • Examples: • DVD’s • Records • Engines • 2000 rpm (revolutions per minute)

  20. Example Problem • An Indy car can complete 120 laps in 1.5 hours on a circular track. Calculate the average angular speed of the Indy car.

  21. Example Problem Answer • 0.14 rad/s

  22. Angular Acceleration • Angular Acceleration – The time of change of angular speed, expressed in radians per second per second.

  23. Angular Acceleration Equation • α= Angular Acceleration (rad/s2) • Lower case Greek letter, “alpha” • Δω= Change in Angular Velocity • t = Time Interval

  24. Example Problem • A yo-yo at rest is sent spinning at an angular speed of 12 rev/s in 0.25 seconds. What is the angular acceleration of the yo-yo?

  25. Example Problem Answer • 301.6 rad/s²

  26. All Points Have Same Speed and Acceleration • If a point on the rim of a bicycle wheel had a greater angular speed than a point closer to the axel, the shape of the wheel would be changing. • Which can not happen in normal everyday riding. • All points on a rotating object have the same angular speed & acceleration.

  27. Comparing Linear and Angular QuantitiesAngular Substitutes for Linear Quantities

  28. Kinematic Equations Can be Used for Circular Motion • The same equations used for linear motion with constant acceleration, can be used for circular motion with constant angular acceleration. • Change the variables, by using the table in the previous slide and you have the new kinematic equations.

  29. Kinematic Equations for Constant Angular AccelerationAngularLinear

  30. Example Problem • A barrel is given a downhill rolling start of 1.5rad/s at the top of a hill. Assume a constant angular acceleration of 2.9rad/s² • If it takes 11.5s to get to the bottom of the hill, what is the final angular speed of the barrel? • What angular displacement does the barrel experience during the 11.5s ride?

  31. Example Problem Answers • 35 rad/s • 210 rad

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