Arc Length and Angular Velocity

1. Arc Length and Angular Velocity Circular Motion and Linear Analogues

2. Recap Yesterday, we verified that the circumference of a circle is the distance travelled in one rotation. That means: We can extend this new knowledge to anything moving in a circle. The distance travelled in one rotation around the axis is equal to the circumference of the path taken… but we need some new terms.

3. Arc length Suppose we take just a segment of the circle (< 1 rotation). What distance is travelled then? • Arc length, s = rθ; r=radius, θ=angle (radians)

4. Period & Frequency • The period, T of an object is the time required to make one full rotation. Units: seconds (s) • The period of the Earth’s rotation about it’s axis is one day (24 hrs = 1440 min = 86400 s) • The frequency, f is the number of rotations every second. Units: cycles per second, or Hertz (Hz) • The Earth rotates once every day, so it’s frequency is • Therefore,

5. Angular Velocity • If the object moves through an angle, θ in t time, t, then the angular velocity, ω is • Angular velocity – the angle an object moves through in a period of time. • Unit: radians per second,

6. Tangential Velocity • The velocity vector is tangent to the circular path at all times. We know this from our “Hit the Target” Activity. • Our velocity vector changes with time as the radial vector changes. This is usually just a directional change (the radial vector points from the center of the circle to the position of the object, so it changes direction as the object moves around the circle.) The further you are from the center, the faster you’re going.

7. Examples

8. Extension • Since an object with angular speed ω , travels 2π radians per rotation, we can use it to find the period by , and since • We can put v in terms of T by solving for ω to get

9. Summary • When objects move in a circle, the circumference of the circle is equal to the distance travelled in one rotation: so for any angle, the distance travelled is • Period is the time for one complete cycle • Frequency is the number of cycles per second • Angular velocity is the angle travelled through per unit time • Tangential velocity is related to the angular velocity by the radius, implying that the further from the center you are, the faster you travel.

10. Example • Suppose that an object executes uniform circular motion, radius r = 1.2m, at a frequency of f = 50 Hz (the object rotates 50 times in one second). The repetition period of the motion is • It’s angular frequency (velocity) is • The tangential velocity is

11. Assignment Create vocab tabs for • Arc length • Period • Frequency • Angular velocity • Tangential (linear) velocity Include: • A definition • The equation • A drawing • Colors

12. Angular Acceleration & Rotational Mechanics

13. Angular Acceleration. • Linear acceleration, a is the rate at which linear velocity, v changes. • Angular acceleration, α is the rate at which angular velocity, ω changes.

14. Example When a bowling ball is first released it slides down the alley before it starts rolling.  If it takes 1.2 s for a bowling ball to attain an angular velocity of 6 rev/sec, determine the average angular acceleration of the bowling ball.

15. Rotational Mechanics Rotational/Angular Linear

16. Example A skater initially turning at 3 rev/sec slows down with constant angular deceleration and stops in 4 seconds.  Find her angular deceleration and the number of revolutions she makes before stopping.

17. Connection between Translational & Rotational Motion • Up to this point we have treated linear and rotational motion separately.  However, there are many cases, such as a ball rolling on the ground, where both rotational and linear motion occurs.  To investigate this relationship, examine the wheel of radius r that has rolled a distance s as shown below.

18. Translational & Rotational • A reference point (A) on to the rim of the wheel has been identified.  We first observe that the arc length traced out by our reference point is equal to the distance s that the arc wheel moves.  This distance is called the tangential distance.  The relation between s and θ has already been used in our definition of radian measure. • Tangential distance: s = rθ We will now use this equation and the defining equation of angular velocity to determine a relationship between linear and rotational velocity.  To do this we consider a wheel rotating with constant angular velocity ω as it moves to the right with linear velocity v.

19. Translational & Rotational • We remember from yesterday: • Finally, let’s examine the relation between linear and rotational acceleration.  If we examine the previous results (s = rθ and vt = ωr) a reasonable guess would be that our derivation will yield at = rα.  If an object is experiencing a constant angular acceleration then both  ω and vt must be changing.  From our definition of constant angular acceleration we have The quantity in parentheses is the rate at which tangential velocity is changing.  This quantity is called the tangential acceleration at.  It is the rate of change in tangential velocity of any point in the rim.  In addition, it is also the linear acceleration of the wheel. 

20. Dimensional Analysis (converting between units) • Write 1000 rpm in rad/sec. • 1 rotation = 2πradians • Write 6000 cm/min in km/hr. • 100,000 cm = 1 km