1. Circular Motion Kinematics 8.01 W04D1

2. Today’s Reading Assignment: W04D1 Young and Freedman: 3.4; 5.4-5.5 Supplementary Notes: Circular Motion Kinematics 2

3. Learning Objectives 1.Calculate time derivative of vector of constant magnitude 2.Recognize that circular motion is a constraint (like sliding down an inclined plane) 3.Calculate direction and magnitude of velocity and acceleration for circular motion when: i) constant speed ii) non-constant speed 4.Explain difference between polar and cartesian coordinates 5.Express position, velocity, and acceleration in polar coordinates 6.Explain meaning of angular velocity, angular frequency, and angular acceleration 7.Know how to use different expressions for radial component of acceleration in terms (v,ω) and then in terms of (r, v), (r,ω), (r, f), and (r,T)

4. Concept Question: Time Derivative of a Vector The time derivative of a vector 1.is always zero. 2.is non-zero if the magnitude of the vector is changing in time. 3.is non-zero if the direction of the vector is changing in time. 4.can be zero if the both it’s magnitude and direction are changing in time in such a way as to add to zero. 5.Two of the above 6.None of the above.

5. Circular Motion Kinematics

6. Concept Question: Time Derivative of Position Vector for Circular Motion A point-like object undergoes circular motion at a constant speed. The vector from the center of the circle to the object 1.has constant magnitude and hence is constant in time. 2.has constant magnitude but is changing direction so is not constant in time. 3.is changing in magnitude and hence is not constant in time.

7. Position and Displacement : position vector of an object moving in a circular orbit of radius R : change in position between time t and time t+  t Position vector is changing in direction not in magnitude. The magnitude of the displacement is the length of the chord of the circle:

8. Review: Definition of Instantaneous Velocity At time t, consider a finite time interval, calculate the displacement. As, generate a sequence of average velocities. The limiting value of this sequence is defined to be the instantaneous velocity at the time t.

9. Direction of Velocity Sequence of chord directions approach direction of velocity as approaches zero. The direction of velocity is perpendicular to the direction of the position and tangent to the circular orbit. Direction of velocity is constantly changing.

10. Small Angle Approximation When the angle is small: Power series expansion Using the small angle approximation with, the magnitude of the displacement is

11. Speed and Angular Speed The speed of the object undergoing circular motion is proportional to the rate of change of the angle with time: Angular speed:

12. Circular Motion: Constant Speed, Period, and Frequency In one period the object travels a distance equal to the circumference: Period: the amount of time to complete one circular orbit of radius R Frequency is the inverse of the period:

13. Table Problem: Demo Rising Ping Pong Ball A wheel is connected via a pulley to a motor. A thread is knotted and placed through a hole in a ping pong ball of mass m. A wing nut secures the thread holding the ball a distance R from the center of the wheel. The wheel is set in motion. When a satisfactory angular speed  is reached, the string is cut and the ball comes off at a tangent to the spinning wheel, traveling vertically upward a distance h. Find the frequency of the rotating wheel just before the string is cut.

14. Demo: Centripetal vs. Centrifugal B102 1.http://tsgphysics.mit.edu/front/index.php?page=de mo.php?letnum=B%20102&show=0http://tsgphysics.mit.edu/front/index.php?page=de mo.php?letnum=B%20102&show=0 A wooden ball is attached to the rim of a spinning wheel. The ball is held in place by a string. When the string is cut, the ball flies in a straight tangent to the wheel.

15. Cylindrical Coordinate System Coordinates Unit vectors

16. Circular Motion: Vector Description Use plane polar coordinates Position Velocity

17. Acceleration and Circular Motion When an object moves in a circular orbit, the direction of the velocity changes and the speed may change as well. For circular motion, the acceleration will always have a radial component (a r ) due to the change in direction of velocity. The acceleration may have a tangential component if the speed changes (a t ). When a t =0, the speed of the object remains constant.

18. Direction of Radial Acceleration: Uniform Circular Motion Sequence of chord directions approaches radial inward direction as approaches zero Perpendicular to the velocity vector Points radially inward

19. Table Problem: Magnitude of Change in Velocity for Circular Motion Find the change in the magnitude of the velocity in terms of v and Δθ. Use the small angle approximation as necessary.

20. Magnitude of Change in Velocity: Circular Motion Solution Change in velocity: Magnitude of change in velocity: Using small angle approximation

21. Radial Acceleration: Constant Speed Circular Motion Any object traveling in a circular orbit with a constant speed is always accelerating towards the center. Direction of velocity is constantly changing. Radial component of a r (minus sign indicates direction of acceleration points towards center)

22. Concept Question A car is rounding a circular turn of radius R=200m at constant speed. The magnitude of its centripetal acceleration is 2 m/s 2. What is the speed of the car? 1.400 m/s 2.20 m/s 3.100 m/s 4.10 m/s 5.None of the above.

23. Alternative Forms of Magnitude of Radial Acceleration Parameters: speed v, angular speed ω, angular frequency f, period T

24. Concept Question: Circular Motion As the object speeds up along the circular path in a counterclockwise direction shown below, its acceleration points: 1.toward the center of the circular path. 2.in a direction tangential to the circular path. 3.outward. 4.none of the above.

25. Tangential Acceleration Circular Motion: Magnitude of Velocity is Non-Constant When the magnitude of the velocity changes by an amount Then the tangential acceleration is the time rate of change of the magnitude of the velocity Angular acceleration: rate of change of angular velocity with time

26. Circular Motion: Vector Description Use plane polar coordinates Position Angular Speed Velocity Angular Acceleration Acceleration

27. Concept Question: Cart in a Turn A golf cart moves around a circular path on a level surface with decreasing speed. Which arrow is closest to the direction of the car’s acceleration while passing the point P?

28. Concept Question: Circular Motion An object moves counter-clockwise along the circular path shown below. As it moves along the path its acceleration vector continuously points toward point S. The object 1.speeds up at P, Q, and R. 2.slows down at P, Q, and R. 3.speeds up at P and slows down at R. 4.slows down at P and speeds up at R. 5.speeds up at Q. 6.slows down at Q. 7.No object can execute such a motion.

29. Next Reading Assignment: W04D2 Young and Freedman: 3.4; 5.4-5.5 Experiment 2: Circular Motion 29