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PH 121 12.1-12.2
Transcript of PH 121 12.1-12.2
To this point we have used a particle model
for our study, for the most part at least.
This chapter branches out a bit and we
start to study rigid bodies, and rotation.
A rigid body is one that doesn't bend,
think of things like a disk, a ball, a gear
We will also play with divers, ice skaters,
and other such people in motion.
Now they aren't strictly rigid, but for
some of their motion we'll pretend.
Rigid bodies have two basic types of motion.
Translational: We have used this to great extent.
Blocks slide down ramps, projectile motion, etc
Rotational motion: we also did this to some extent.
Disks spinning, objects turning.
In addition to these two types a combination can be made
This hammer has a translational motion
as well as a rotational motion.
maybe easier to see is this wrench. It is sliding across a desk
in a straight line, but it is also rotating.
For a bit of review we have rotational quantities
and their relationships
With sign convention as well
Further review, we have rotational kinematic equations
Finally the connection between rotational
and translational motion.
A high speed drill reaches 2000 rpm in 0.5 s
What is the drill's angular acceleration?
Through how many revolutions does it turn during this first 0.5 s?
As we begin to talk about rigid body
rotation, we need to talk about what
the body rotates around.
this is the center of mass.
You can see this in the previous pictures.
Here is another example that illustrates this motion.
What then is the center of mass?
Be careful it is not just the center of the object
it is the mass weighted center.
Meaning that you find where all the mass
of the object is and then find the center of that.
So, how do you do that?
How can you find the center of mass of an object?
Here are three methods
1. Spin the object.
It will rotate around the center of mass.
think back to the youtube video.
2. Hang the object by 2 or 3 different
locations, and draw a line straight down
from the fixed point.
The intersection of the lines will be the
center of mass.
3. Use an equation
This equation assumes that you can add up every single
particle and its mass at its location. in practicality , not good.
Your book then goes into a bit of calculus and
takes the limit of this summation to an integral
and gives you the integral form of the sum.
This is still not a practical equation.
You would need to know an equation describing
the mass density of an object. As well as a perfect
size or dimension of an object.
This would work great in uniform rods, and circles,
but for any engineering purpose, no good.
Also note that in the current form this integral
is not helpful, you need a dx or a dy.
For a uniform rod you could consider
that the ratio of mass and distance over the total
mass and distance are the same.
This would give:
The three masses shown are connected by massless rigid rods. What are the coordinates of the center of mass?
Questions from your studying
If I throw a hammer, or tennis racket, and asked you to find the velocity of any point on it at any time,
could you do it?
An 18 cm long bicycle crank arm, with a pedal at one end is attached to a 20 cm diameter sprocket, the toothed disk around which the chain moves. A cyclist riding this bike increases her pedaling rate from 60 rpm to 90 rpm in 10 s.
A. What is the tangential acceleration of the pedal?
B. What length of chain passes over the top of the sprocket during this interval?
A 100 g ball and a 200 g ball are connected by a 30 cm long massless rigid rod. The balls rotate about their center of mass at 120 rpm. What is the translational speed of the 100 g ball?
"Why don't radians have units attached to them? And why haven't we created a unit that could be attached to it?"
"How are radial acceleration and tangential velocity related? Why are Radial acceleration and radial velocity not related?"
"Why is it that rotational motion is not rotation about the center of mass?"
"Can we go over rotation about the center of mass and equation 12.4?"