PES 1110 Fall 2013, Spendier Lecture 26/Page 1

Today: - Practice angular Kinematic problems (10.4) - Relating the Linear and Angular Variable (10.5) Review last Friday Angular Motion A rotating object has infinitely many linear speeds but only one angular speed. Arclength and angle Measure for rotational distance s = r θ (θ is in radians) 2π rad = 360º = 1 rev (revolution is also a unit of angle)

Units: 1 No Unit!s mr m

q Angels have no units, they are unitless.

The angle θ is the key quantity since all points rotate through the same angle! To locate a point on a spinning object we describe the point by θ. We must distinguish linear motion = distance/time from angular motion = angle/time Angular Velocity

Average angular speed : 2 1

2 1 minrad rev or avg RPM

t t t sq q qw

(Instantaneous) angular speed: 0

limt

dt dtq qw

RHR - Curl the fingers of your right hand in the “sense" (clockwise or counterclockwise) of the rotation. Your extended thumb, points in direction of w

(point = arrow tip coming at you) (feathers = arrow flying away from you)

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Angular Acceleration

Average angular acceleration: 2 12

2 1

rad avg t t t sw w wa

,

(Instantaneous) angular acceleration: 0

limt

dt dtw wa

Example 1: The angular position of a point on the rim of a rotating wheel is given by θ(t) = 4.0 t - 3.0 t2 + t3, where θ(t) is in radians and t is in seconds. a) What is the average angular acceleration for the time interval that begins at t = 2.0 s and ends at t = 4.0 s? b) What is the instantaneous angular acceleration at the end of this time interval?

PES 1110 Fall 2013, Spendier Lecture 26/Page 3

Angular Kinematics Since the angular motion is so similar in mathematical description to linear motion, we expect that there will be an equivalent number of equations describing angular motion for constant angular acceleration α. What should these equations look like? Consider x θ v ω a α So we just write the 5 linear equations with angular variables: 1) 0(t) = + tw w a

2) 20 0

1( )2

t t t

3) 2 20 02w w a q q

4) 20

1( )2

t t t

5) 0 01( )2

t t

This is a short way to find these equations. We could have also used the methods we applied in lecture 3 to derive these equations. For example using calculus we can proof the first equation from above

By definition: ddtwa

dt da w ( )

0 0 (0)

tt t

dt dt dw

w

a a w since α = constant

0( ) (0) ( )t t ta w w w w

0( ) t tw w a

PES 1110 Fall 2013, Spendier Lecture 26/Page 4

Example 2: A merry-go-round rotates from rest with an angular acceleration of 1.50 rad/s2. How long does it take to rotate through the first 2.00 rev?

Relating Linear and Angular Velocity We already know that a spinning object has infinitely many values of linear velocities when it spins. We can calculate the linear velocity for any given point. Start with a point at θ = 0 and rotate up to some angle θ. The arclength s is the linear distance traveled on this circle form the initial to the final point. This is a distance in meters.

We can take this distance and change it into a velocity:

dsvdt

This gives me the linear speed of this rotating point. Angular velocity is changing angle:

ddtqw

Arclength and radius are all related as long as I use angle in units of radians: s = r θ (take time derivative on both sides)

= ds d dr r v rdt dt dt

qq w (since r is constant)

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v rw The linear speed v at any point is the radius (out to this point) times angular speed ω. ( Note: you need to pick the correct radius. The point on the rotating object might not be on the edge) Example 3 A wheel is spinning at 45RPM. What is the linear speed of the point which is 0.25m away from the center?

2 1 min45 0.25 1.18 /min 1 60rev radv r m m s

rev spw

When relating linear and angular quantities you MUST use radians! Example 4: A wheel is rotating about an axis that is in the z-direction. The angular velocity ωz is - 6.00 rad/s at t = 0 s, increases linearly with time, and is +8.00 rad/s at t=7.00 s. We have taken counterclockwise rotation to be positive. a) Is the angular acceleration during this time interval positive or negative? b) Calculate ωz(t) and plot ωz(t). c) During what time interval is the speed of the wheel increasing? Decreasing?

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Connected Rotating Objects What happens when we have two rotating objects we connect together? For example, when they are connected by a nonslipping chain or belt (like in your bicycle), the two rotating objects must have the same linear speed: v1 = v2 since v rw , the two rotating objects must have different angular speed if 1 2r r : ω1 r1 = ω2 r2

This also applies to two gears in contact. Example 5: With what angular speed must gear A of radius 24 mm rotate if we wish gear B of radius 100 mm (bottom) to rotate at 100RPM?

ωA rA = ωB rB

100 100400

25B B

AA

mm RPMr RPMr mmww

The important thing here is the ratio of the distance of your gears. For example, in the gear in the back of your bike is smaller. So when you pedal, you are setting the ω for the big gear (the front one). So the rear gear will spin with a ω that is larger. That is why you can bike very fast. But we know the smaller the rear gear is the more effort we have to put in – more power.