- Good morning. Today we are going to compare
linear and rotational values as a part
of our review of the AP Physics C Mechanics curriculum.
♫ Flipping Physics ♫
- Billy, what are the linear and rotational
equations for displacement?
- Well, displacement is change in position,
which equals position final minus position initial.
For linear position the symbol
is sometimes X, it could be Y, Z, or sometimes R.
For angular position, the symbol is Theta.
Oh, and displacement is a vector.
- Bobby, how about the equations
for average and instantaneous velocity?
- Average velocity equals change
in position over change in time
and instantaneous velocity equals
the derivative of position with respect to time.
Velocity is a vector, we use the symbols
for linear and angular position Billy just identified,
and the symbol for angular velocity
is a lower case Omega which looks
like a curly 'W'.
- Bo, please give me the equations
for average and instantaneous acceleration.
- It is almost exactly what Bobby just said
only replace position with velocity.
Acceleration is a vector,
and the symbol for angular acceleration
is Alpha however some people like
to call it ...
- Fishy thing!
- Now let's talk about the uniformly accelerated motion
or UAM equations.
I've written out the four UAM equations
in a form that's not quite the same
as what's written on your AP equation sheet.
For example, the first equation
is velocity final equals velocity initial
plus acceleration times time,
where that time is defined as time final
when time initial is equal to zero.
I find this form is just a little bit easier
to work with that what is on your AP equation sheet.
- Okay, now let's move on to talking about
the uniformly angularly accelerated motion,
or U fishy M equations.
- [All] U fishy M?
- The angular U fishy M equations
follow the same form as the linear UAM equations.
For example, the second U fishy M equation is
angular position final equals angular position initial
plus angular velocity initial times time
plus one half times the angular acceleration
times time, squared.
- Billy, what is the linear equation for Mass?
- What?
- There is no linear equation for mass.
- Right
- Oh, okay.
- Exactly. Now, Billy, what is
the rotational equation for mass?
- Rotational mass is called Moment of Inertia.
The symbol for Moment of Inertia
is capital I and there are two equations.
The Moment of Inertia of a system of particles
is the sum of the mass of each particle
times r squared, or the square of the distance
each particle is from the axis of rotation,
and the Moment of Inertia for a rigid object
with shape is the integral of r squared
with respect to mass.
- Bobby, please give us the equations
for linear and rotational kinetic energy.
- Linear or translational kinetic energy
equals one half mass times velocity squared,
and rotational kinetic energy equals
one half mass times velocity squared,
only the mass is rotational mass or moment of inertia,
and the velocity is angular velocity.
- Billy, how about Newton's Second Law?
- Net force equals mass times acceleration
where force and acceleration are vectors.
There is also net force equals
the derivative of momentum with respect to time
where momentum is a vector.
Uh, then there is the rotational form
of Newton's Second Law, net torque equals
moment of inertia times angular acceleration
and net torque also equals the derivative
of angular momentum with respect to time,
and torque, fishy thing, and angular momentum
are all vectors.
- Bo, please give us the equations for force and torque.
- The equation for force ... uh
- There is no general equation for force,
there are only equations for net force,
which I just talked about.
- Right, there are equations for some forces
like the force of gravity and the force of friction,
however there is no general equation
for a single force.
- The equation for torque is torque equals
the cross product of r and f.
The symbol for torque is the lower case Greek letter Tau,
r is the position vector from the axis of rotation
to the location of the force,
and torque, r, and F are all vectors,
and yes, the order matters because
the cross product is not commutative.
So it has to be r cross F, in that order.
- Bobby, the equations for power, please.
- Linear or translational power
is the derivative of work with the respect of time,
which also equals the dot product
of force and velocity where
both force and velocity are vectors.
Uh, I guess the equation is initially the same
for rotational power, rotational power
equals the derivative of work with respect to time
only now it's the dot product of torque
and angular velocity where torque
and angular velocity are vectors.
- And lastly, Billy, please tell us about momentum.
- Linear momentum, the symbol for which is
a lower case p, not an upper case P which is for power,
so linear momentum equals mass times velocity
where momentum and velocity are vectors.
There are two equations for angular momentum,
the symbol for which is an upper case L,
the angular momentum of a particle
equals r cross p, where r is the distance from
the center of mass in the particle
to the axis of rotation and p is the momentum,
and the angular moment of a rigid object with shape
equals the object's moment of inertia
times the object's angular velocity.
Oh, and angular momentum, r, momentum,
and angular velocity are all vectors.
- Thank you everybody. That completes our review
of linear and rotational values
as a part of the AP Physics C Mechanic curriculum.
Next, feel free to enjoy my Universal Gravitation Review
or visit my AP Physics C review webpage.
Thank you very much for learning with me today,
I enjoyed learning with you.