Download H Ch 7 Notes - Angular Motion.notebook

H Ch 7 Notes ­ Angular Motion.notebook
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Honors Physics
One of the first ideas we learned about this year was displacement...we've always talked about linear displacement. (Dx) This measures how far an object moves. Lesson 7.1
Dx = xf ­ xi This chapter we'll be talking about angular displacement. (Dq) This measures how far an object spins
Angular Velocity
Jan 23­12:58 PM
Jan 22­9:46 AM
Note that degrees is an arbitrary measure of angle. Since radians are fundamentally connected to circles, we will measure angle in radians. This allows us to easily "jump" between linear and angular quantities:
1 rotation =
Dx = Dqr
Dq = Dqf - Dqi
Now, average (linear) velocity (vavg) is a measure of an object's average displacement per unit time. In circular motion, linear velocity is sometimes called tangential velocity because its direction is tangent to the circle.
Similary, we can calculate an object's average angular velocity. (ωavg)
360o =
2π rads
and a simple substitution gives us the following "jumper" which is true for both average and instantaneous velocities.
v = ωr
Jan 22­9:58 AM
Jan 22­10:01 AM
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H Ch 7 Notes ­ Angular Motion.notebook
Notice these angular quantities can be either positive or negative, indicating that they can be thought of as having direction...they are vectors! To accurately do this, we would use what is called "The right hand rule" which assigns to every rotation, a vector which points along the axis of rotation for which the object spins counter­clockwise. If you take AP, we'll discuss this more, but for the purposes of this class, we'll simply use:
Example 1:
A DVD spins in the player at up to 1600 rpms.
a. Find the angular velocity of the disc. (168 rad/s)
Counterclockwise = positive
b. If there is a scratch 5 cm from the center of the disc, find its velocity. (8.38 m/s)
Clockwise = negative
Note thought that this is flawed because, like right and left, clockwise and counterclockwise are relative and differ from one observer to the next. c. How far will the scratch move in 45 seconds? (33.5 m)
d. Through what angle will the disc turn in 45 seconds? (670 rads = 107 rot)
Jan 23­1:15 PM
Example 2.
A bowling ball rolls at 18 mph. If the ball has a radius of 11 cm then (assuming the ball doesn't slide at all)..
a. Find the ball's angular velocity. (73.1 rad/s)
b. How many times will the ball roll as it goes down the 60 foot lane? (26.5 rot)
Jan 22­10:18 AM
Honors Physics
Lesson 7.2
Angular Acceleratoin
Jan 24­8:34 AM
Jan 23­12:58 PM
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H Ch 7 Notes ­ Angular Motion.notebook
One of the next ideas we learned about this year was (instantaneous) velocity...we've always talked about linear velocity (v) which measures how fast an object is moving (in m/s) at a given moment. In circular motion, linear velocity is sometimes called tangential velocity because its direction is tangent to the circle. Now, (linear) acceleration (a) is a measure of how quickly an object's velocity is changing. In circular motion this is sometimes called tangential acceleration.
This chapter we'll be talking about (instantaneous) angular velocity (ω) which measures how fast an object is spinning (in rad/s) at a given moment. Similary, we can calculate an object's angular acceleration (α)
and a simple substitution gives us the following "jumper" .
a = αr
Jan 22­9:46 AM
Using the jumper equations, we can easily see that each of our constant linear acceleration equations gives a corresponding constant angular acceleration equation.
Jan 22­10:01 AM
Examples
The carousel at Comerica Park has a radius of 3.5 m. The ride accelerates, from rest, and takes 4.5 seconds to reach a maximum frequency of 0.05 Hz.
a. Find the ride's angular acceleration. (0.070 rad/s2)
b. Through what angle will the ride turn in the time it takes to achieve its highest speed? (0.707 rad)
c. What tangential velocity will a rider at the outer edge of the ride reach? (1.10 m/s)
Jan 22­10:18 AM
Jan 22­10:18 AM
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H Ch 7 Notes ­ Angular Motion.notebook
Example 2
A motorcycle driving at 60 mph slows and stops in 8 seconds. If the tires have a radius of 0.35 m a. then how many times will the tires rotate before the bike stops? (48.8)
b. find the angular acceleration of the tires. (­3.35 rad/s2)
Honors Physics
Lesson 7.3a
Torque & Equilibrium
Jan 22­10:18 AM
So we have learned that (net) force causes acceleration. Similarly, (net) torque causes angular acceleration. Torque (τ) is a measure of how much a force causes an object to rotate.
Jan 23­12:58 PM
So let's calculate some torques:
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0°
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The torque produced by a force is directly proportional to two values:
• The force (F) applying the torque.
• The distance (d) between the pivot point and the point where the force is being applied. This is called the "lever arm." Jan 23­1:04 PM
Jan 23­3:10 PM
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H Ch 7 Notes ­ Angular Motion.notebook
Eventually, we will learn how these torques determine the object's angular acceleration. But, first, we will limit our discussion of torque to objects which are in "static equilibrium." This simply means that the object has a net force and a net torque of zero. We'll start with the classic "teeter­
totter" example. c. Now...let's do this. Find the net torque around the point where Sally is sitting.
Suppose a teeter­totter has a (massless, for now) board which is 4.0 meters long. The fulcrum is at the center. Sally, who has a mass of 60 kg sits at one end of the board. d. Now...let's do this. Find the net torque around the point where Roy is sitting.
a. Where must Roy, who has a mass of 75 kg, sit to balance the board?
b. Make a FBD of the board.
Jan 23­1:20 PM
Honors Physics
Jan 23­3:20 PM
Okay...let's try something else. A 20 kg board is 4.0 meters long. The board is sitting on a support, 0.75 m from one end. Roy stands at the short end of the board and holds it level. a. How much force does Roy exert, and in which direction?
Lesson 7.3b
b. How much force is exerted by the support?
To solve this problem, we must first learn about "Center of Mass."
Torque, Equilibrium & Center of Mass
Jan 23­12:58 PM
Jan 23­3:22 PM
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H Ch 7 Notes ­ Angular Motion.notebook
Center of Mass (com) is an object's average position of mass. Usually in physics, we think of objects as particles...but real objects are not "particles." BUT, the com of an object is important because
• Forces applied to an object at its com will cause the object to move exactly as if it was a particle (ie, with no additional rotation)
• Gravity can always be treated as a force which acts at the com. Click here
Honors Physics
Lesson 7.4
And Click here
Soooooooo, we can apply this to the board problem on the previous slide!
Jan 23­3:31 PM
Rotational Inertia & Angular Acceleratoin
Jan 23­12:58 PM
Sooooo, we've learned that and object's linear acceleration depends on:
• The net force on the object ( ΣF )
• The object's mass (inertia) ( m )
Similarly, an object's angular acceleration depends on:
• The net torque on the object ( Στ )
• The object's rotational inertia (aka "moment of inertia") ( I )
In the same way that inertia (mass) is a measure of an object's resistance to a change in motion, rotational inertia is a measure of an object's resistance to a change in angular motion. An object's rotational inertia depends on:
• The mass of the object
• The size & shape of the object (ie, its dimensions)
• The axis of rotation.
Jan 23­3:50 PM
Jan 23­4:05 PM
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H Ch 7 Notes ­ Angular Motion.notebook
So, Newton's 2nd Law tells us:
ΣF = ma
And the angular counterpart to this would be:
Example 1
If a car's tires have radius of 33 cm and an inflated mass of 10 kg, then find the net torque needed to accelerate the car from 0 mph to 60 mph in 6 seconds. You may treat the tire as a solid cylinder. (7.37 Nm)
Στ = Iα
Jan 23­4:05 PM
Jan 23­4:28 PM
Example 2
Roy is attempting to lift a large piece of plywood which is sitting on the floor.
The uniform board is 1.5 m wide, and 3.0 m long, and has a mass of 60 kg. a. He grabs the short side of the board, and lifts up with a force of 300 N. What angular acceleration will this give the board initially? (0.10 rad/s2)
b. What if he grabs the long side of the board and lifts with 300 N? (0.20 rad/s2)
Jan 24­12:15 PM
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