Advanced Physics, Chapter 8 (8-1 only), “Describing Rotational Motion”, pp. 197-152
and Chapter 6 (6-2) Circular Motion, pp. 153-156: Syllabus
Remember that you can access the online physics textbook
in pdf format at the following web address:
http://www.glencoe.com/ose/
click link and type " A7AFD238AE " under student access
code
There will be two online homework assignments:
• Homework 021 (rotational kinematics)
• Homework 022 (centripetal forces)
One test at the end of the unit
Helpful websites:
http://dev.physicslab.org/Document.aspx?doctype=3&filename=RotaryMotion_Rotatio
nalKinematics.xml
http://hypertextbook.com/physics/mechanics/rotational-kinematics/
http://media.pearsoncmg.com/bc/aw_young_physics_11/pt1a/Media/RotMotionStatics/
RotationalKinematics/Main.html
http://www.knowsoft.com/HS_physics/pdf/Rotational_Kinematics.pdf
http://hyperphysics.phy-astr.gsu.edu/Hbase/circ.html#rotcon
http://www.physicsclassroom.com/Class/circles/U6L1c.cfm
Rotational Motion and Centripetal Acceleration
Chapter 8 (Section 8-1)
Chapter 6 (Section 6-2)
Two general types of circular motion:
a) Rotation
b) Revolution
Two different ways to discuss motion for a rotating object:
a) angular displacement
rotating objects ‘sweeps’ out an angle
measure angle (θ) usually in radians
θ in radians = arc length (s) / radius (r)
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 1
11/1/2009
θ: radians ↔ degrees ↔ rotations or revolutions
The time it takes to complete one orbit or one complete circle? Called the Period of
rotation (units of sec).
example:
the reciprocal of Period?
Called frequency
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 2
11/1/2009
angular velocity (ω): This is a measure of how much angle is being swept out by a
rotating object in a given time
ω=
∆θ radians
∆t
sec
Other units to specify ω
If we have an object rotating at a constant angular velocity:
∆θ = ω ( ∆t )
Notice its similarity to: Δx = v (t)
But often we don’t have objects rotating constantly. We can object speed up or slow
down their rotational velocities. In linear systems, we call a change of velocity an
acceleration, but in rotating systems, we call this acceleration an angular acceleration
and it is given the symbol, α
α=
∆ω
∆t
radians
sec
sec
You know what? You can do the exact same derivations we did with translational
kinematics to find rotational kinematics equations if we are considering a case with
constant angular acceleration. Basically, what comes out is essentially the same
equations but with different variables:
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 3
11/1/2009
Rotational Kinematics equations
Comparing linear motion with angular motion:
Constant velocity linear motion:
5
4
3
2
1
0
x (displacement, meters)
x f = xi
0 1 2 3 4 5
+ v(t)
t (time seconds)
Constant angular velocity
5
4
3
2
1
0
(θ, angular displacement, radians)
0 1 2 3 4 5
t (time seconds)
Equation:
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 4
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Constant linear acceleration:
vf = vi + at
vf 2 = vi2 + 2a(∆x)
xf = xi + vit + ½at2
Constant angular acceleration
(ω, angular velocity, rads/sec)
(θ, angular displacement, rads)
t (time, seconds)
(ω, angular velocity, rads/sec)
t (time, seconds)
(θ, angular displacement, rads)
t (time, seconds)
t (time, seconds)
Equations
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 5
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Sample Problems:
1. Some of the motors we used in our fan cars were rated at
10,200 rpm. If this velocity stays constant, calculate the
angular velocity in rad/s. How if takes the fan car 5 seconds
to complete the race, how many radians would have been
swept out in this time? How many rotations would the
propeller have undergone in these 5 seconds?
(answers = 1068.14 rads/sec; 5340.7 rads; 850 rotations)
2. The blades of a fan running at low speed turn at 250 rpm. When the fan is switched
to high speed, the rotation rate increases uniformly to 350 rpm in 5.75 seconds. What is
the angular acceleration of the blades during this time? How many revolutions do the
blades go through while the fun is accelerating?
(answers: 1.82 rads/s^2; 28.7 or 28.8 revolutions)
3. A CD player reaches its operational rotational rate of 3.5 rps
after making 2 rotations. What is its angular acceleration in
rads/s2?
(answer = 19.2 rads/s/s)
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 6
11/1/2009
Connecting linear and angular quantities…
I had mentioned on page 1 that there is another way to describe motion of a rotating
object. This is called tangential velocity (vt)
Why called tangential?
For one complete rotation,
Is there a relationship between angular velocity (ω) and tangential velocity (vt)?
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 7
11/1/2009
Other ancillary notes of Chapter 6:
What gives objects accelerations?
What give objects angular acceleration?
The measure of a force's tendency to produce a rotation about
an axis is called torque. That is, if a force is used to begin to
spin something, or to attempt to spin something, a torque is
generated. A torque would also be generated if a force was
used to stop something from spinning.
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 8
11/1/2009
Situations dealing with torque…
Balanced torques: See-Saw problems…
Equilibrium…
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 9
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Angular Momentum and its conservation
We know from our unit on momentum that momentum is conserved if we
neglect any external forces such as friction. (collision problems)
We see something similar with objects rotating…
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 10
11/1/2009
Chapter 6 (Section 6-2) Uniform Circular Motion
Special type of acceleration for objects moving along a circular path:
- let’s go back to the definition of acceleration
- Could acceleration exist even if an object is moving at constant velocity?
liquid accelerometer demo
Acceleration is directed inward…we call it a centripetal acceleration (ac)
- This acceleration is not caused by the change of the magnitude of the velocity
vector but because the direction of the velocity vector is changing.
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 11
11/1/2009
The centripetal acceleration expression is obtained from analysis of constant speed
circular motion by the use of similar triangles. From the ratio of the sides of the
triangles:
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 12
11/1/2009
Two important relationships to guide our thinking:
(a) Newton’s 2nd Law: Fnet = ma
Forces causing objects to move in a circle are called centripetal forces…
(b) Fnet = Fc =
mv 2
r
What forces act centripetally? What forces tend to keep objects moving in circular
paths??
a) tension
b) gravity
c) friction
d) normal forces
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 13
11/1/2009
Sample Problems
1. A 900-kg car makes a 180-degree turn with a speed of 10.0 m/s.
The radius of the circle through which the car is turning is 25.0 m.
Determine the force of friction and the coefficient of friction acting
upon the car. (answer: the force of friction is 3600 N; the coefficient of friction of 0.41.)
2. The coefficient of friction acting upon a 900-kg car is 0.850. The car is making a 180degree turn around a curve with a radius of 35.0 m. Determine the maximum speed with
which the car can make the turn. (answer = maximum speed of 17.1 m/s.)
3. A 1.5-kg bucket of water is tied by a rope and whirled in
a circle with a radius of 1.0 m. The bucket of water travels at
a constant speed of 4 m/s. Determine the acceleration, the
values of the tension at the top and the bottom of the circular
path.
(answers: Tension at the top = 9.3 N; Tension at the bottom = 38.7 N)
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 14
11/1/2009
4. Two identical objects are placed on a flat turntable that can rotate at different
velocities. One of the objects is placed 25 cm away from the center while the other is
placed twice this distance. The coefficient of static friction (μ) for each object is 0.5.
The turntable’s rotational speed (vt) is gradually increased. At what tangential velocity
will each object just begin to slide off (maximum tangential velocity)?
(answers = 1.11 m/s for the inner object and 1.56 m/s for the outer object)
5. In 1901, circus performer, Allo “Dare Devil” Diavolo
introduced the student of riding a bicycle in a loop-the-loop.
If the radius of the loop was 3 m, what is the least tangential
velocity Diavolo could have at the top of the loop to remain
in contact with the surface?
(answer = 5.4 m/s is the least tangential velocity)
6. How many rpm would a 7.5 m radius Ferris wheel need to make for the passengers to
feel “weightless” at the topmost point? What would the period of revolution for the
passengers?
(answers: rpm = 10.9 rpm; Period (T) = 5.5 seconds/revolution)
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 15
11/1/2009
How do objects continue moving in circles if the centripetal force is directed inward?
If not enough vt…
If not enough Fc…
Via the 1st law, object moves off tangentially…
Reference frames and circular motion
a can…
From a reference point outside the can…
From a reference point inside the can…
‘Centrifugal’
Simulated gravity
Centrifugal force can feel like gravity
Space stations
Artificial gravity
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 16
11/1/2009
Different visions of space stations
Adv. Physics, Rotational Kinematics/Centripetal Forces Notes, p. 17
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