**Equilibrium, Rolling,**

Vectors and Momentum

Vectors and Momentum

**Equilibrium**

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

advantage.

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.

**Rolling motion**

**Rolling motion**

is caused by torque, in most cases it is

from friction.

Here we will note the motion and constraints.

is caused by torque, in most cases it is

from friction.

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:

Or...

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

origin

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

variables.

Angular momentum

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

quantities.

Demo

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?

22.3 rpm

50 rpm

"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?"