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PH 121 12.8-12.11
Transcript of PH 121 12.8-12.11
Vectors and Momentum
Recall that we now have a version of
Newton's law in rotational as well as
translational reference frames.
When we have a rigid body in
equilibrium BOTH sets work to our
In solving these problems we:
1. Note all the forces (FBD)
2. Pick any ANY axis for rotation
3. Use equations:
Example, biology student walks
to the end of a beam. Physics
student gets there.
is caused by torque, in most cases it is
Here we will note the motion and constraints.
With rolling motion we need
both the translational and
rotational kinetic energy.
All rotating objects have a
moment of inertia.
And they all come in the
generic form of mr^2
If we choose to use a constant
'c' we can write any Moment as:
With that, and also relation of angular
to linear velocity, as shown here:
We can rewrite our Kinetic energy.
Substitute in for moment of inertia (I) and
angular velcity (omega), gives:
This is our simplified version of kinetic energy of an
object that is translating and rotating.
Down hill race.
Two objects with the same mass and radius both begin with the same initial velocity at the bottom of an inclined plane. If one is a solid sphere, and other is a hollow shell sphere. Which one travels further up the hill
A. solid (c=2/5)
B. shell (c=2/3)
C. neither, g is fixed.
Note we did this before as an example of
moment of inertia.
Let us now predict a final velocity of a
solid sphere, disk, and hoop.
Initially I only have potential energy
due to gravity.
Finally (if we ignore drags) we only
have kinetic energy, So...
Solving for Velocity
Note there is no
Mass or radius
Rather only the shape (c)
Vectors in rotational motion
Right hand rule:
We use the right hand rule to give the direction
of Omega and Alpha.
We said that they were '+' CCW, and '-' CW
In truth the direction points along the axis
following the right hand rule.
We can now revisit the idea of Torque
How to calculate a Cross product...
Also we can define the Angular Momentum
of a particle as it moves, measured from the
Doing some calculus, or maybe review of previous
material we have a relation between the change of
momentum and force.
This is true in rotational frames, just with Rotational
As must mentioned we can relate radial distance
and classic momentum to get
an angular momentum.
In addition we can compare to the classical derivation. (p=mv)
This angular momentum is a
conserved quantity here as
well, just like linear momentum.
Follow the same steps as before
find initial momentum
Find final momentum
equate the two and solve for unknown.
Also as we have been going we have made
conversions between rotational and translational
here is a bit of a list.
Here is one such table showing these comparison
A merry-go-round is a common piece of playground equipment. A 3.0 m diameter merry go round with a mass of 250 kg is spinning at 20 rpm. John runs tangent to the merry go round at 5.0 m/s, in the same direction that it is turning, and jumps onto the outer edge. John's mass is 30 kg. What is the merry go round's angular velocity, in rpm, after John jumps on?
A 200 g, 40 cm diameter turntable rotates on frictionless bearings at 60 rpm. A 20 g block sits at the center of the turntable. A compressed spring shoots the block radially outward along a frictionless groove in the surface of the turntable. What is the turntable's rotation angular velocity when the block reaches the outer edge?
"I'm still having trouble with what exactly we're supposed to do with torques like with the problem yesterday with the pulleys. what am i setting equal to what?"
"Which type of object would get to the bottom of a hill first, if all were release on same hill at same height and had same mass and radius?"
"Could you go over example 12.15?"
"How do rolling objects have both translational and rotational kinetic energy?"
"Why is the motion of the center of mass equal to the tangential motion at the edge?"
"Could you show in class how to solve a problem about kinetic energy of a rolling object?"