1. Chapter 8
Rotational Motion

2. Warm-up
What is the difference between circular motion, rotational motion, and translational motion?
What is a rigid body?
Identify 5 angular quantities and the formula in solving it.
How do you change a radian to degree and vice versa?

3. Rotational Motion
Rigid body = a body with a definite shape that doesn’t change, so that the particle composing it stay in fixed positions relative to one another.

4. Angular Quantities
In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is r. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined:
where s is the arc length.

5. Angular Quantities Angular displacement:
The average angular velocity is defined as the total angular displacement divided by time:

6. Angular Quantities
The angular acceleration is the rate at which the angular velocity changes with time:

7. Angular Quantities
Every point on a rotating body has an angular velocity ω and a linear velocity v.
They are related:

8. Angular Quantities
Therefore, objects farther from the axis of rotation will move faster.

9. Angular Quantities
If the angular velocity of a rotating object changes, it has a tangential acceleration:
Even if the angular velocity is constant, each point on the object has a centripetal acceleration:

10. Angular Quantities
Here is the correspondence between linear and rotational quantities:

11. Angular Quantities
The frequency is the number of complete revolutions per second:
Frequencies are measured in hertz.
The period is the time one revolution takes:

12. Constant Angular Acceleration
The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.

13. Consider a disk rotating on a stationary rod.
If we now view this disk from the top,

14. s = rθ θ = s/r s = rθ 2πr = rθ θ = 2π radians
We see that when the disk rotates so that the arc length (s) equals the length of the radius of the disk (r), the subtended central angle (θ) will equal 1 radian. The resulting equation is
where the unit of a radian represents the dimensionless measure of the ratio of the circle's arc length to its radius
s = rθ
θ = s/r
If the disk rotates through one complete revolution, then s equals the entire circumference and θ equals 2π radians
Since one complete revolution equals 360º, we now have the conversion that
360º = 2π radians 1 radian = 180/π or approximately 57.3º
s = rθ 2πr = rθ θ = 2π radians

15. Problem 1 An old phonograph record revolves at 45 rpm.
What is its angular velocity in rad/sec?
Once the motor is turned off, it takes 0.75 seconds to come to a stop. What is its average angular acceleration?
ω = 45 rev/min = 45 (2π/60 radians/sec) = 4.71 rad/sec
givens:
ωf = 0, ωo = 4.71 rad/sec, t = 0.75 seconds.
using the equation
ωf = ωo + αt
we can determine that
α = ( )/0.75 = rad/sec2

16. How many revolutions did it make while coming to a stop?
using the equation
θ = ½(ωf + ωo)t
we can determine that
θ = ½ ( ) θ = 1.77 radians
since there are 2π radians in every revolution,
θ = rev.

17. a. How many revolutions does the blade require to alter its speed?
A fan that is turning at 10 rev/min speeds up to 25 rev/min in 10 seconds.
b. If the tip of one blade is 30 cm from the center, what is the final tangential velocity of the tip?
a. How many revolutions does the blade require to alter its speed?
using the equation
v = rω
allows us to determine that  
v = (0.30)(2.63) v = m/sec
using the equation
θ = ½(ωf + ωo)t
we can determine that
θ = ½( )10 θ = 18.4 radians
since there are 2π radians in every revolution, θ = 2.92 rev

18. Consider a standard analog wall clock with a second hand, minute hand, and hour hand
a. Calculate the angular velocity of the second hand of a clock
ω = 1 rev/min = (2π/60 radians/sec) = rad/sec
b. If the second hand is 8" long (there are 2.54 cm in every inch), what is the linear velocity of the tip of the second hand?
using the equation
v = rω
allows us to determine that
v = (0.203)(0.105) v = m/sec

19. Two wheels are connected by a common cord
Two wheels are connected by a common cord. One wheel has a radius of 30 cm, the other has a radius of 10 cm
When the small wheel is revolving at 10 rev/min, how fast is the larger wheel rotating?
Since the two wheels share the same tangential velocity, their angular velocities will be inversely proportional to their radii.
vlarge=vsmall
rlargeωlarge= rsmallωsmall
ωlarge = (rsmall/rlarge) ωsmall
ωlarge = (0.10/0.30)(10) = 3.33 rev/min

20. A rotor turning at 1200 rev/min has a diameter of 5 cm
A rotor turning at 1200 rev/min has a diameter of 5 cm. As it turns, a string is to be wound onto its rim
  since the wheel is turning at a constant angular velocity, we can use the equations
θ = ωt
s = rθ
substituting gives us the equation
s = r(ωt)
and we can calculate the amount of string wrapped around the exterior of the rotor
s = (0.025)(1200)(0.105)(10)
s = 31.5 meters
How long a piece string will be wrapped in 10 seconds?