1. Unit 5: Conservation of Angular Momentum
Chapter 10: Rotational Motion About a Fixed Axis
Chapter 11: General Rotation
Chapter 12: Static Equilibrium
With this information you’ll have a good working knowledge of the mechanics of translation and rotational motion!
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2. Review: Angular Quantities
There is a close parallelism between the variables of linear motion and those of angular motion.
The motion of a rigid body can be described with both translation motion and rotational motion.
Consider the disk at the right undergoing purely rotational motion, that is all points move in a circle about the axis of rotation, which is projecting from the screen.
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3. We use R rather than r to indicate distance from the axis of rotation.
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4. Indicating Angular Position of a Point
Given by an angle with respect to an axis.
A point P “moves through” angle q as it travels along arc l.
Angles can be given in degrees or more conveniently in radians.
One radian is the angle subtended by an arc equal to the radius.
Note that one radian is the same angle for any sized circle.
1 rad
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5. More on Radians By definition then q = l/R
where R is the radius of a circle,
and l is the arc length subtended by q.
Note radians are dimensionless!
Radians are easily related to degrees since the 360o arc length of a complete circle is 2pR:
360o = l/R =2pR/R = 2p rads  1 rad = 57.3o
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6. Angular Variables of Motion: w and a
Consider the wheel, it’s angular displacement after a bit of rotation is given by:
In complete analogy with average velocity the average angular velocity, w, is defined as:
And the instantaneous angular velocity is
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7. The units of w are rad/s and for a they are rad/s2.
We can also define average and instantaneous angular acceleration in analogy to linear acceleration:
The units of w are rad/s and for a they are rad/s2.
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8. The Relationship between Angular and Linear Velocity
Each point on a rotating rigid body has nonzero w and v.
The figure helps to under-stand the relationship between the two for P.
The magnitude of the linear velocity is given by
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9. Note that different radii have equal angular velocity but
v=Rw
Note that different radii have
equal angular velocity but
very different linear velocity
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10. The Relationship between Angular and Linear Acceleration
If an object’s angular velocity changes there will also be angular acceleration.
Every point on the object will then undergo tangential acceleration.
But also recall there is a radial acceleration:
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11. Collecting Results
We can also write the angular velocity, w, in terms of the frequency, f.
Since
A frequency of 1 rev/sec = an angular velocity of 2p rads/sec, we can say:
f = w/2p or w=2pf
The unit of frequency rev/s is given the name hertz(Hz) and since revolutions are not a true unit (just a place keeper) 1Hz=1s-1.
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12. Equations of Motion for Rotational Motion
The definitions of average and instantaneous angular velocity and angular acceleration are identical to linear velocity and acceleration except for a variable change:
Recall the definitions of average and instantaneous velocity and acceleration led to the four equations of linear motion for constant acceleration.
An identical analysis for angular motion at constant angular acceleration would lead to the same four equations with the replacement:
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13. Note since the equations are identical there is no need for
a re-derivation, this is a
pretty common technique!
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14. Rolling Motion
One of the most familiar combined translational and rotational motions, balls, bicycles, cars…
Rolling without slipping depends on static friction since the point of contact is momentarily at rest.
The figure shows a wheel rolling to the right.
From the earth’s reference from the wheel’s CM (the axle) undergoes translational motion v.
From the axle’s point of view point P is undergoing rotational motion only with velocity –v.
In this frame, since there is only rotational, motion v=Rw.
So the overall situation has
Translation motion of v.
Rotational motion v=Rw.
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15. Example: The Motion of a Bicycle.
A bicycle with wheels of diameter 0.680m slows down from vo=8.40m/s to rest over a distance of 115m.
What is the
Initial angular velocity
Total number of revolutions of each wheel
Angular acceleration of the wheel
Time to stop.
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16. Angular Velocity: This is no different than the first wheel we considered:
Revolutions: In coming to a stop each point must have traveled 115 m, thus
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17. Rotational Dynamics
As we discussed linear motion we moved from kinematic equations to dynamics. The study of forces led to conservation of energy and momentum.
We take the same approach in Chapters 10 and 11 and study the dynamics of rotational motion until we ultimately determine the contribution of rotational motion to the concepts of energy conservation and momentum conservation.
In fact, we’ll determine a new conservation law: the conservation of angular momentum.
Rotational motion is, in a sense, richer than linear motion, because the precise location of a force on an object determines the rotational motion. For instance if a force is applied at the edge of a rigid object, it causes greater rotational motion than if it’s applied near the center. In order to deal with this we need to develop a new concept called torque.
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18. An Intuitive Definition of Torque
From Newton’s 2nd Law a net force causes acceleration.
Net Force  Acceleration
or a net external force causes linear motion to change.
Likewise, something external must also cause rotational motion to change. A force isn’t enough: if we push on a propeller blade at its very center no additional rotational motion occurs. However, if we push at the edge of the propeller the rotational motion does change. So to change the rotational motional a force AND its position are important
Net Force at the Proper Position Rotational Acceleration
This off-center force is actually called a torque or the moment of the force about the axis.
Note the analog between force which changes a and torque which changes a.
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19. Torque to distance between axis and force.
As with most concepts in mechanics you already have a pretty good intuitive idea of torque: it’s the degree to which you can set an object rotating by pushing it with a force.
Consider opening a door: You know that it opens more quickly the harder you push. So it’s easy to see that torque is proportional to force:
Torque applied force.
But for the door there’s more, if we push near the hinges or the axis of rotation hardly anything happens. If we push near the doorknob we get a much greater response. So the torque or the ability to rotate the door is proportional to the distance between the axis and the point of the application of the force:
Torque to distance between axis and force.
Even more, if you push toward or away from the hinges nothing happens, one must push in a direction perpendicular to the hinges,
Torque depends on the direction of the force.
In summary then the ability to set an object rotating or the torque depends on the magnitude and direction of the applied force as well as the distance from the axis of rotation
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20. The line of action is an extended line drawn co-linear with the force.
To go further let’s consider a door and apply a force that lies in a plane perpendicular to the axis of rotation. (In the figure the axis is perpendicular to the page and the force lies in the plane of the page.)
The drawing shows two new concepts: the line of action and the lever arm of the force on the door.
The line of action is an extended line drawn co-linear with the force.
The level arm is the distance between the line of action and the axis of rotation. The lever arm is measured on a line perpendicular to both
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21. With these new concepts we can see that our intuitive ideas on torque are met if torque equals the magnitude of the force times the lever arm:
if force is greater the torque is greater
if the distance between the axis and force increases the torque increases
if the force points towards the axis no torque is present.
Definition of Torque:
Magnitude: Torque = (Lever Arm) x (Magnitude of the force) or in symbols:
Direction: The torque is positive if the force tends to produce a counterclockwise rotation about the axis, and negative if the force tends to produce a clockwise rotation:
SI Units : meter-Newton (mN)
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22. Example: The Door
Problem: A force of 55N is applied to a door. The lever arms in the three parts of the drawing are a) 0.80 m, b) 0.60m, and c) 0 m. Find the torque in each case.
Answer: =
(0.80m)(55N) = 44 mN
(0.60m)(55N) = 33 mN
(0.00m)(55N) = 0 mN
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23. Alternate Expressions
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24. An Example: A Tendon
Consider the ankle joint at point A and Achilles tendon attached to the heel at point P. The tendon exerts a force of magnitude F=720 N as shown. Determine the torque of this force about the ankle joint located m from point P.
Answer:
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