1. Describing and Dynamics
Rotational Motion
Describing and Dynamics

2. Rotational Motion Describing Rotational Motion Grad = revolution
Fractions of revolution measured in grads, degrees, or radians
Grad = revolution
Degree = revolution
Radian = revolution

3. Rotational Motion Angular Displacement
Theta, θ, represents angle of revolution
Counterclockwise rotation positive, clockwise negative
Change in angle = angular displacement
d = rθ
Displacement (d) = rotation through angle,
θ, at distance, r, from center

4. Rotational Motion Angular Velocity
Velocity = displacement divided by time
Angular velocity is angular displacement divided by time required to make displacement
Angular velocity is represented by the Greek letter omega, ω

5. Rotational Motion Angular Velocity
If velocity changes over time, average velocity not equal to instantaneous velocity at any given instant
Angular velocity = average angular velocity over a time interval, t
Instantaneous angular velocity = slope of graph of angular position versus time
Measured in rad/s
So, for Earth, ωE = (2π rad)/[(24.0 h)(3600 s/h)] = 7.27×10─5 rad/s

6. Rotational Motion Angular Velocity v = rω
Counterclockwise rotation also results in positive angular velocity
If angular velocity is ω, then linear velocity of point at distance, r, from axis of rotation given by
Speed at which object on Earth’s equator moves due to Earth’s rotation is v = r ω = (6.38×106 m) (7.27×10─5 rad/s) = 464 m/s
v = rω

7. Rotational Motion Angular Acceleration
Change in angular velocity divided by time required to make that change
Measured in rad/s2
If △𝑣 positive, then ω also positive
Angular acceleration also average angular acceleration over time interval Δt

8. Rotational Motion Angular Acceleration a = r
Find instantaneous angular acceleration by finding slope of graph of angular velocity as function of time
Linear acceleration of point at r from axis of object with angular acceleration, α, given by
a = r

9. Rotational Motion Angular Acceleration
A summary of linear and angular relationships

10. Rotational Motion Angular Frequency
Number of complete revolutions made by object in 1s called angular frequency
Angular frequency, f, is given by the equation:

11. Rotational Motion Describing Rotational Motion
Jupiter, the largest planet in our solar system, rotates around its own axis in 9.84 h. The diameter of Jupiter is 1.43 x 108 m.
What is the angular speed of Jupiter’s rotation in rad/s?
What is the linear speed of a point on Jupiter’s equator, due to Jupiter’s rotation?
A computer disk drive optimizes the data transfer rate by rotating the disk at a constant angular speed of 34.1 rad/s while being read. When the computer is searching for a file, the disk spins for s.
What is the angular displacement of the disk during this time?
Through how many revolutions does the disk turn during this time?

12. Rotational Motion Describing Rotational Motion
A cyclist wants to complete 10.0 laps around a circular track 1.0 km in diameter in exactly 1.0 h. At what linear velocity must this cyclist ride?
A 75.0 g mass is attached to a 1.0 m length of string and whirled around in the air at a rate of 4.0 rev/s when the string breaks.
What is the breaking force of the string?
What was the linear velocity of the mass as soon as the string broke?

13. Rotational Motion Rotational Dynamics
Change in angular velocity depends on magnitude of force, distance from axis to point where force exerted, and direction of force
To open door, you exert force. Doorknob near outer edge of door. Exert force on doorknob at right angles to door, away from hinges
To get most effect from least force,
exert force as far from axis of
rotation as possible

14. Rotational Motion Rotational Dynamics
Magnitude of force, distance from axis to point where force exerted, and direction of force determine change in angular velocity
Change in angular velocity depends on lever arm, perpendicular distance from axis of rotation to point where force exerted

15. Rotational Motion Rotational Dynamics
For door, distance from the hinges to point where force exerted
If force perpendicular to radius of rotation then lever arm is distance from axis, r. If force not exerted perpendicular to radius, lever arm reduced

16. Rotational Motion Rotational Dynamics
Lever arm, L, calculated by equation, L = r sin θ,
θ = angle between force and radius from axis of rotation to point where force applied
Torque measures how effectively force causes
rotation. Measured in newton-meters (N·m)

17. Rotational Motion
Rotational Dynamics

18. Rotational Motion Rotational Dynamics
In order for a bolt to be tightened, a torque of 45.0 N•m is needed. You use a m long wrench, and you exert a maximum force of 189 N. What is the smallest angle, with respect to the wrench, at which you can exert this force and still tighten the bolt?
Chloe, whose mass is 56 kg, sits 1.2 m from the center of a seesaw. Josh, whose mass is 84 kg, wants to balance Chloe. Where on the seesaw should Josh sit?

19. Rotational Motion The Moment of Inertia
Equal to mass of object times square
of object’s distance from axis of rotation
Resistance to changes in rotational
motion
Represented by symbol I and has units
of mass times square of distance

20. Rotational Motion Newton’s Second Law for Rotational Motion
Angular acceleration directly proportional to net torque and inversely proportional to moment of inertia
Changes in torque, or moment of inertia, affect rate of rotation

21. Rotational Motion Newton’s Second Law for Rotational Motion
A fisherman starts his outboard motor by pulling on a rope wrapped around the outer rim of a flywheel. The flywheel is a solid cylinder with a mass of 9.5 kg and a diameter of 15 cm. The flywheel starts from rest and after 12 s, it rotates at 51 rad/s.
What torque does the fisherman apply to the flywheel (α= τ/I)?
How much force does the fisherman need to exert on the rope to apply this torque?

22. Rotational Motion The Center of Mass
The center of mass of an object is the point on the object that moves in the same way that a point particle would move
The path of the center of mass of the object below is a straight line

23. Rotational Motion The Center of Mass
To locate the center of mass of an object, suspend the object from any point
When the object stops swinging, the center of mass is along the vertical line drawn from the suspension point
Draw the line, and then suspend the object from another point. Again, the center of mass must be below this point
Draw a second vertical line. The center of mass is at the point where the two lines cross
A wrench, racket, and all other freely-rotating objects, rotate about an axis that goes through their center of mass

24. Rotational Motion
The Center of Mass

25. Rotational Motion The Center of Mass of a Human Body
The center of mass of a person varies with posture
For a person standing with his or her arms hanging straight down, the center of mass is a few centimeters below the navel, midway between the front and back of the person’s body

26. Rotational Motion The Center of Mass of a Human Body
When the arms are raised, as in ballet, the center of mass rises by 6 to10 cm
By raising her arms and legs while in the air, as shown below, a ballet dancer moves her center of mass closer to her head
The path of the center of
mass is a parabola, so the
dancer’s head stays at
almost the same height
for a surprisingly long
time

27. Rotational Motion
Center of Mass and Stability

28. Rotational Motion Center of Mass and Stability
An object is said to be stable if an external force is required to tip it
The object is stable as long as the direction of the torque due to its weight, τw tends to keep it upright. This occurs as long as the object’s center of mass lies above its base
To tip the object over, you must rotate its center of mass around the axis of rotation until it is no longer above the base of the object
To rotate the object, you must lift its center of mass. The broader the base, the more stable the object is

29. Rotational Motion Center of Mass and Stability
If the center of mass is outside the base of an object, it is unstable and will roll over without additional torque
If the center of mass is above the base of the object, it is stable
If the base of the object is very narrow and the center of mass is high, then the object is stable, but the slightest force will cause it to tip over

30. Rotational Motion Conditions for Equilibrium
An object is said to be in static equilibrium if both its velocity and angular velocity are zero or constant
First, it must be in translational equilibrium; that is, the net force exerted on the object must be zero
Second, it must be in rotational equilibrium; that is, the net torque exerted on the object must be zero

31. Rotational Motion Rotating Frames of Reference
Newton’s laws are valid only in inertial or nonaccelerated frames
Newton’s laws would not apply in rotating frames of reference, as they are accelerated frames
Motion in a rotating reference frame is important to us because Earth rotates
The effects of the rotation of Earth are too small to be noticed in the classroom or lab, but they are significant influences on the motion of the atmosphere and therefore on climate and weather

32. Rotational Motion Centrifugal “Force”
An observer on a rotating frame, sees an object attached to a spring on the platform
He thinks that some force toward the outside of the platform is pulling on the object
Centrifugal “force” is an apparent force that seems to be acting on an object when that object is kept on a rotating platform

33. Rotational Motion Centrifugal “Force”
As the platform rotates, an observer on the ground sees things differently
This observer sees the object moving in a circle
The object accelerates toward the center because of the force of the spring
The acceleration is centripetal acceleration and is given by

34. Rotational Motion Centrifugal “Force”
It also can be written in terms of angular velocity, as:
Centripetal acceleration is proportional to the distance from the axis of rotation and depends on the square of the angular velocity

35. Rotational Motion The Coriolis “Force”
A person standing at the center of a rotating disk throws a ball toward the edge of the disk. An observer standing outside the disk sees the ball travel in a straight line at a constant speed toward the edge of the disk

36. Rotational Motion The Coriolis “Force”
An observer stationed on the disk and rotating with it sees the ball follow a curved path at a constant speed
A force seems to be acting to deflect the ball

37. Rotational Motion The Coriolis “Force”
An apparent force that seems to cause deflection to an object in a horizontal motion when the observer is in a rotating frame of reference is known as the Coriolis “force”
It seems to exist because we observe a deflection in horizontal motion when we are in a rotating frame of reference

38. Rotational Motion The Coriolis “Force”
An observer on Earth, sees the Coriolis “force” cause a projectile fired due north to deflect to the right of the intended target
The direction of winds around high- and low-pressure areas results from the Coriolis “force.” Winds flow from areas of high to low pressure

39. Rotational Motion The Coriolis “Force”
Due to the Coriolis “force” in the northern hemisphere, winds from the south blow east of low-pressure areas
Winds from the north, however, end up west of low-pressure areas
Therefore, winds rotate counterclockwise around low-pressure areas in the northern hemisphere
In the southern hemisphere however, winds rotate clockwise around low-pressure areas