Lecture 7B
"The greatness of a nation and its moral progress can be judged by the way its animals are treated.”
--Mohandas Gandhi

Quiz 4
Quiz 4 will include all of Chapter 6.
My suggestion is to do the homework, class examples, and book examples.
Don’t forget your quiz code number and a
Scantron for Friday or you will be penalized.

Chapter 7

Rotational Variables
Up until now, we have only dealt with bodies in translation, which is motion along a line.
Now, we will deal with the consequences of rotational motion.
We need to define useful variables for rotational motion as we did with translational motion.
Angular position is measured by θ [in radians].

Rotational Variables
Angular displacement is the change in angular position:
Δθ   =   θ f
θ i
For ease, when dealing with angular displacement we say that we are dealing with a rigid body.
This means that each point on the object undergoes the same angular displacement.
The convention for angular displacement is that a clockwise displacement is negative and a counterclockwise displacement is positive.
Also, be careful, Δθ can be greater than 2 π rad.

Rotational Variables
Average angular speed is given by:
It is angular displacement over a time interval.
The units of angular speed are [rad/s].
Angular speed will be positive if θ is increasing (ccw).
Angular speed will be negative if θ is decreasing
(cw).

Rotational Variables
Average angular acceleration is given by:
It is the change in angular speed over a time interval.
The units of angular acceleration are [rad/s 2 ].
These variables correspond to translational variables.
θ --> x ω --> v α --> a

Rotational Variables
We can use the constant acceleration equations with rotational variables:
Remember (just like linear motion) these equations rely on the fact that angular acceleration remains constant over the time period in question.

Relating Variables
Sometimes you would like to relate linear variables and rotational variables (and vice versa).
Let’s say you were traveling on the edge of a merry-go-round of radius, r. If you wanted to know your distance travelled, Δ s, if you have rotated through an angular displacement, Δθ , then use:
Δ s   =   r ( Δθ ) [ Δθ in radians]
The bigger the radius, the more distance you have covered per angular displacement.

Relating Variables
Now, let’s say you wanted to know how fast you would travel linearly is you let go of the merrygo-round if your angular velocity was ω ; then use: where v t
is known as tangential speed.

Maintaining circular motion
We know that a body moves in a straight line at constant speed if no net force is applied.
=> A body moving in a circle must experience a net force, even if it moves at constant speed around that circle !!!
Let’s examine that force and acceleration.

Maintaining circular motion
Without a net force the body would fly off tangentially:
There thus must be a force and corresponding acceleration radially inwards to stay on the circle !!!
a c
We call this “centripetal” acceleration.
v t

Two accelerations
We need to distinguish angular acceleration ( α ) and centripetal (also called “radial”) acceleration.
α results in a change of the tangential speed according to: a t
=   α r a c is what’s required to stay on the circle.
The total acceleration is the vector sum of the two !!!
a r a a t

Centripetal Acceleration
Looking at Newton’s 2nd Law: where F c
is the net centripetal force.
Centripetal force will always point toward the center of the circular motion.
In the case of a ball on a string, the tension in the string causes the net force (centripetal force).
In the case of the Moon orbiting the Earth,
Earth’s gravitational force causes the net force
(centripetal force).

In class question
Andrea and Chuck are riding on a merry-go-round.
Andrea rides on a horse at the outer rim of the circular platform, twice as far from the center of the circular platform as Chuck, who rides on an inner horse. When the merry-go-round is rotating at a constant angular speed, Andrea’s tangential speed is:
A) a quarter of Chuck’s.
B) half of Chuck’s.
C) the same as Chuck’s.
D) twice Chuck’s.
E) four times Chuck’s.

In class question
An object moves in a horizontal circular path with a constant speed of 10m/s. Which of the following statements is true concerning the object?
A) Its velocity vector is constant, but its acceleration vector is changing.
B) Its acceleration vector is constant, but its velocity vector is changing.
C) Both its velocity vector and acceleration vector are changing.
D) Both its velocity vector and acceleration vector remain constant.

Example
Centripetal Force
A 1,000kg car rounds a curve on a flat road of radius 50.0m at a speed of 50.0km/hr (14.0m/s).
Will the car make the turn if: a) the pavement is dry and the coefficient of static friction is 0.800?
b) the pavement is icy and the coefficient of static friction is 0.200?
(Note: use max static friction here for the extreme case of the tires almost slipping.)
Answer
First, you must define a coordinate system.
Let’s say that up as +y and call in toward the center of the curve as +x.

Centripetal Force
Answer
Next, we draw a free-body diagram (assume car moves into the board and +x is to the right):
F normal, ground on car
F friction, ground on car car
F gravity, Earth on car
What does that friction force have to be in order for the car to make the turn?
<- this is the force needed to make the turn.

Centripetal Force
Answer
Next, look at the forces in the y-direction:
∑ F y
=   F
N
–   F g
=   0
F
N
=   F g
=   mg   = (1000kg)(9.80N/kg)   =   9800N
For part a)
F fr,max
=   μ s
( F
N
)   =   (0.800)(9800N)   =   7840N
This is more than the 3920N necessary to make the turn.
For part b)
F fr,max
=   μ s
( F
N
)   =   (0.200)(9800N)   =   1960N
This is less than the 3920N necessary to make the turn.

Centripetal Force
To solve this problem with icy roads, curves are banked.
Here the normal force adds to the overall centripetal force.
In the picture we see that nsin θ will provide the centripetal force and ncos θ will balance out the gravitational force.

Centripetal Force
Dividing these two equations gives us:
The greater the angle of the bank, the more velocity and/or the smaller the radius of curvature you can have and still make the turn.

For Next Time (FNT)

Rotational Variables
Example
A wheel has a constant angular acceleration of
3.0rad/s 2 . During a certain 4.0 second interval, you measured that the wheel had an angular displacement of 120 radians. If the wheel started at rest, how long had it been in motion at the start of the 4 second interval?
Answer
First, you must define a coordinate system.
Let’s say that the wheel starts moving counterclockwise in the positive direction.

Rotational Variables
Answer
Let’s list the quantities we know:
ω o
= 0 <-- it starts from rest
α = +3.0rad/s 2
ω <-- don’t know
Δθ <-- don’t know t <-- finding
But we also know that in a 4sec. span it went through an angle of 120rad.
We can use the third equation to find the angular velocity at the beginning of that time span.

Rotational Variables
Answer
Use:
Thus, it started out the 4 sec. interval with an angular velocity of 24rad/s.
We have a new known: ω   =   24rad/s.

Rotational Variables
Answer
Since Δθ is missing we can use:
The only difference between translational kinematic equations and rotational kinematic equations is that as the object rotates it will gain angular displacement even as it keeps repeating the same position.
Angular displacement is cumulative.