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ROTATIONAL MOTION and the LAW of GRAVITY

ROTATIONAL MOTION and the LAW of GRAVITY. Ch 7:. Measuring Rotational Motion. ROTATIONAL MOTION : Motion of a body that spins about an axis. Ex: Ferris Wheel Any point on a Ferris wheel that spins about a fixed axis undergoes circular motion.

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ROTATIONAL MOTION and the LAW of GRAVITY

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  1. ROTATIONAL MOTION and the LAW of GRAVITY Ch 7:

  2. Measuring Rotational Motion • ROTATIONAL MOTION: Motion of a body that spins about an axis. • Ex: Ferris Wheel • Any point on a Ferris wheel that spins about a fixed axis undergoes circular motion. • To analyze the motion is it convenient to set up a fixed reference line. And assume that at time t=0, a cart is on the reference line. Reference line

  3. After time interval t2 the cart advances to a new position. • In this time interval, the line from the center to the cart moved through angle theta with respect to the reference line. • Likewise, the cart moved a distance s, measured from the circumference of the circle; s is arc length. Cart (at time t2) s Reference line

  4. Theta and Arc Distance (s) • If the cart moves through an angle twice as large, How would the new distance compare with the distance shown here represented by s? Cart (at time t2) s Reference line

  5. Theta and Arc Distance (s) • If the cart moves through an angle twice as large, How would the new distance compare with the distance shown here represented by s? • THE NEW DISTANCE WOULD BE TWICE AS LARGE! Cart (at time t2) s Reference line

  6. Angles can be measured in RADIANS • ***ALMOST ALL EQUATIONS IN THIS CHAPTER AND THE NEXT REQUIRE THAT ANGLES BE MEASURED IN RADIANS*** • Radian  an angle whose arc length is equal to the radius, which is approximately equal to 57.3 degrees.

  7. Angles in RADIANS… • Any angle theta measured in radians is defined by: Theta = s (arc length) r (radius) Radian is a pure number, with NO dimensions. Units cancel, and we use the abbreviation rad to identify unit.

  8. Circumference… • When the cart of the Ferris wheel from our previous example goes through an angle of 360 degrees. The arc length s is equal to the circumference of the circle, or 2 π r • Thus, theta = s = 2πr= 2πrad r r

  9. Any angle in degrees can be converted to an angle in radians by multiplying the angle measured in degrees by: 2π=π 360 degrees 180 degrees

  10. Angular motion is measured in units of radians. Because there are 2πradians in a full circle, radians are often expressed as a multiple of π.

  11. Look at Figure 7-3 in your text on page 245 • Determine the radian measure equivalent to 75 degrees.

  12. Look at Figure 7-3 in your text on page 245 • Determine the radian measure equivalent to 75 degrees. _5 π 12

  13. ANGULAR DISPLACEMENT • Describes how much an object has rotated • Book Definition: the angle through which a point, line, or body is rotated in a specified direction and about a specified axis. • EQUATION: Angular displacement (in radians)= Change in arc length Distance from axis OR  θ = s/r

  14. EXAMPLE: • Earth has an equatorial radius of approximately 6380 km and rotates 360 degrees every 24 hours. • What is the angular displacement (in degrees) of a person standing at the equator for 1 hour?

  15. EXAMPLE: • Earth has an equatorial radius of approximately 6380 km and rotates 360 degrees every 24 hours. • What is the angular displacement (in degrees) of a person standing at the equator for 1 hour?  θ= 360/24 = 15 degrees

  16. EXAMPLE: • Earth has an equatorial radius of approximately 6380 km and rotates 360 degrees every 24 hours. • Convert the angular displacement (in degrees) to radians.

  17. EXAMPLE: • Earth has an equatorial radius of approximately 6380 km and rotates 360 degrees every 24 hours. • Convert the angular displacement (in degrees) to radians.  θ(rad) = π /180 (θdeg) =π /180 (15 deg)  θ(rad) = 0.26 rad

  18. EXAMPLE: • Earth has an equatorial radius of approximately 6380 km and rotates 360 degrees every 24 hours. • What is the arc length traveled by this person?

  19. EXAMPLE: • Earth has an equatorial radius of approximately 6380 km and rotates 360 degrees every 24 hours. • What is the angular displacement (in degrees) of a person standing at the equator for 1 hour?  θ = s/r (r) θ = s (6380) (0.26) = 1658.8 km ~1700km

  20. Angular Speed… • Describes rate of rotation • The average angular speed, θ avg, is the ratio between the angular displacement,  θ, to the time interval, t, that the object takes to undergo that displacement. θ avg=  θ/ t Units are given in rad/s

  21. Example • An Indy car can complete 120 laps in 1.5 hours. Even though the track is an oval rather than a circle, you can still find the angular speed. Calculate the average angular speed of the Indy car.

  22. Example • An Indy car can complete 120 laps in 1.5 hours. Even though the track is an oval rather than a circle, you can still find the angular speed. Calculate the average angular speed of the Indy car. 1: Convert hours to seconds. 2: Convert total angular displacement from degrees to radians 3: Solve for average angular velocity. 1.5 hrs = 1.5 * 60 * 60 = 5400 s  θ= 360 deg * 120 = 43200 degrees = 43200 (π /180) = 753.98 radians avg =  θ / t avg = 753.98 rad / 5400 s avg = 0.1396 = 0.14 rad/s

  23. Angular Acceleration • Occurs when angular speed changes • It is the time rate of change of angular speed, expressed in radians per second per second (think about linear acceleration units…m/s/s or m/s2) • Equation: • avg= θ 2 - θ 1=  θ= change in angular speed t2 - t1t time interval

  24. Example… • A top that is spinning at 15 rev/s spins for 55 s before coming to a stop. What is the average acceleration of the top while it is slowing?

  25. Example… • A top that is spinning at 15 rev/s spins for 55 s before coming to a stop. What is the average acceleration of the top while it is slowing? • avg= θ2 - θ 1=  θ= change in angular speed • t2 - t1t time interval • avg= 15 (360) = 0 - 5400=98.18 (π /180) = • 55 55 • - 1.7 rad/s/s

  26. HOMEWORK TONIGHT… • PAGE 247 # 1 and 3 • Page 248 # 1 • Page 250 # 1 and 2 • READ 7-1 and 7-2

  27. Q.O.T.D. 12-1-2009 • Convert the following from degrees to radians: a. 17 deg b. 50 deg c. 170 deg d. 270 deg • What is the ratio that defines the average angular speed? • When a wheel rotates about a fixed axis, do all points on the wheel have the same angular speed?

  28. QOTD- ANSWERS 1 a. 0.297 rad b. 0.873 rad c. 2.967 rad d. 4.712 rad 2- wavg = change in angular displacement time interval. delta theata/delta time (rad/s) 3- YES. Otherwise, the wheel would change shape!

  29. ROTATIONAL MOTIONTangential and Centripetal Acceleration Ch 7: Section 2

  30. Tangential Speed • The instantaneous linear speed of an object along the tangent to the object’s circular path. • Objects in circular motion have a tangential speed.

  31. Tangential Speed • A point on an object rotating about a fixed axis has tangential speed related to the object’s angular speed. • When the object’s angular acceleration changes, the tangential acceleration of a point on the object changes.

  32. Tangential Speed Which cloud would have to move with a faster tangential speed? Compare their angular speeds.

  33. Tangential Speed Which cloud would have to move with a faster tangential speed? Compare their angular speeds. The cloud furthest from the center/axis has the larger tangential speed. Their angular speeds are exactly the same. See fig 7-6 Page 253

  34. How do you calculate Tangential Speed? • Remember that… • Angular displacement = change in arc length/radius or delta theta = delta s/r Average angular speed = angular displacement/time or wavg = delta theta/delta t Vt = r ω Tangential Speed = radius x angular speed

  35. Tangential Speed • A golfer has a max angular speed of 6.3 rad/s for her swing. She can choose between two drivers, one placing the club head 1.9 m from her axis of rotation, and the other placing it 1.7 m from the axis. • 1. Calculate the tangential speed of the club head for each driver. • 2. All other factors being equal, which driver is likely to hit the ball farther?

  36. Tangential Speed • A golfer has a max angular speed of 6.3 rad/s for her swing. She can choose between two drivers, one placing the club head 1.9 m from her axis of rotation, and the other placing it 1.7 m from the axis. • 1. Calculate the tangential speed of the club head for each driver. • Vt = rω = 1.9 (6.3) = 11.97 ~ 12 m/s • Vt = rω= 1.7 (6.3) = 10.71 ~ 11 m/s

  37. Tangential Speed • A golfer has a max angular speed of 6.3 rad/s for her swing. She can choose between two drivers, one placing the club head 1.9 m from her axis of rotation, and the other placing it 1.7 m from the axis. • 2. All other factors being equal, which driver is likely to hit the ball farther? • The longer driver will hit the ball further because its club head has a higher tangential speed.

  38. TANGENTIAL SPEED EXAMPLE… • A softball pitcher throws a ball with a tangential speed of 6.93 m/s. If the pitcher’s are is 0.66m long, what is the angular speed of the ball before the pitcher releases it?

  39. TANGENTIAL SPEED EXAMPLE… • A softball pitcher throws a ball with a tangential speed of 6.93 m/s. If the pitcher’s are is 0.66m long, what is the angular speed of the ball before the pitcher releases it? • Vt = r ω • 6.93 = 0.66 (ω) • 6.93/0.66 = ω • W = 10.5 rad/s

  40. Tangential Speed • A point on an object rotating about a fixed axis has tangential speed related to the object’s angular speed. • When the object’s angular acceleration changes, the tangential acceleration of a point on the object changes.

  41. Tangential Acceleration • Tangential Acceleration is tangent to a CIRCULAR path • It is the instantaneous linear acceleration of an objected directed along the tangent to the object’s circular path. • EQUATION for Tangential Acceleration: at= r α Tangential acceleration = radius x angular acceleration

  42. Example… • A yo-yo has a tangential acceleration of 0.98m/s/s when it is released. The string is wound around a central shaft of radius 0.35 cm. What is the angular acceleration of the yo-yo?

  43. Example… • A yo-yo has a tangential acceleration of 0.98m/s/s when it is released. The string is wound around a central shaft of radius 0.35 cm. What is the angular acceleration of the yo-yo? 0.35cm = 0.0035 m at= r α 0.98 = 0.0035 (α) 0.98/.0035 = α 280 rad2 = α

  44. Centripetal Acceleration • The acceleration directed toward the center of a circular path. • An acceleration due to a change in direction-even when an object is moving at a constant speed) Given by…. Ac = vt2/r Can also be found using the angular speed… Ac = ω2 x r

  45. UNIFORM CIRCULAR MOTION • Uniform circular motion occurs when an acceleration of constant magnitude is perpendicular to the tangential velocity.

  46. Tangential vs. Centripetal Acceleration • Centripetal and tangential acceleration are NOT the same. • The tangential component of acceleration is due to changing speed • The centripetal component of acceleration is due to changing direction.

  47. Review Questions… • Describe the path of a moving body whose acceleration is constant in magnitude at all times and is perpendicular to the velocity. • An object moves in a circular path with a constant speed, v. Is the objects velocity constant? Is its acceleration constant? Explain. • Give an example of a situation in which an automobile driver can have a centripetal acceleration but no tangential acceleration.

  48. Describe the path of a moving body whose acceleration is constant in magnitude at all times and is perpendicular to the velocity. A CIRCLE. • An object moves in a circular path with a constant speed, v. Is the objects velocity constant? Is its acceleration constant? Explain. no. direction is changing. no. the direction of ac is changing, at = 0 • Give an example of a situation in which an automobile driver can have a centripetal acceleration but no tangential acceleration. A car driving in a circle at a Constant Speed.

  49. KEY TERMS TO KNOW FOR YOUR TEST FRIDAY… • Angular Acceleration • Angular Displacement • Angular Speed • Centripetal Acceleration • Gravitational Force • Radian • Rotational Motion • Tangential Acceleration • Tangential Speed

  50. Demo links • http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/VertCircular/VertCircular.html • http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/RTZCoordSystem/RTZCoordSystem.html • http://w3.shorecrest.org/~Lisa_Peck/Physics/syllabus/mechanics/circularmotion/hewitt/Source_Files/08_CentripetalForce_VID.mov

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