mr.p: Good morning. Today we are going to review
the rotational dynamics topics covered
on the AP Physics 1 exam.
Bo: Hey, guys.
Billy: Hey, Bo.
Bobboy: Hi, Bo.
♪ Flipping Physics ♪
mr.p: Let's start with torque.
The symbol for torque is a lowercase tau
and torque is the ability to cause
an angular acceleration of an object.
The equation for torque is the lever arm,
or moment arm multiplied by the force
and we're going to talk about that equation
in just a little bit.
But, before we do, I want to say that
everything you ever needed to know about torque
you already know, because you have opened many doors.
This is the top view of a door, and this is the hinge,
or what we call the axis of rotation..
And, you have learned from experience
that the best way to open a door
is to apply a force far from the hinge
or far from the axis of rotation.
In other words, if you apply a force at the hinge
at the axis of rotation, you will not rotate the door,
and you have learned if you apply
a force perpendicular to the door
the farther you get from the axis of rotation
the easier it is to open the door. In physics terms,
this is because the farther you apply
that perpendicular force to the door
from the axis of rotation, the larger the r perpendicular
the moment arm, the lever arm.
Therefore, a larger torque.
Therefore, a larger angular acceleration.
You also know that the most efficient angle
to push on the door with is at 90 degrees to the door.
Any other angle...
will reduce the moment arm because the sine of theta
decreases as you move away from 90 degrees.
When you apply the force at a 90 degree angle to the door,
the sine of 90 degrees equals one.
Then therefore the moment arm is equal to r
which is the maximum value for the moment arm.
Torque is a vector,
and I wish the AP Physics 1 curriculum
had the right hand rule in it
to describe the direction of torque.
However, it doesn't, so we will use
clockwise and counterclockwise
to describe the direction of torque.
Bo: Who knew? AP Physics 1 is too cool for the right hand rule.
Billy: Makes me sad.
mr.p: We also need the dimensions for torque. Bo?
Bo: Moment arm is in meters times force in Newtons
so Newton meters. Isn't that joules?
mr.p: Yes. Torque is in Newton meters which is joules.
However, because torque and energy are very different things,
torque is a vector; energy is a scalar for example.
We leave torque in Newton meters to identify the difference.
Now, let's talk about the rotational form
of Newton's second law.
The linear form of Newton's second law
is that the net force equals mass times acceleration
where both force and acceleration are vectors.
The rotational form of Newton's second law
is that the net torque is equal to I times
the angular acceleration where torque
and angular acceleration are both vectors.
That means that this I, or what's called
moment of inertia, is rotational mass.
In other words, the moment of inertia or rotational mass
is a measure of an object's resistance
to angular acceleration.
The moment of inertia for a system of particles
is the sum of the mass of each particle
times the square of the distance each particle
is from the axis of rotation for that system of particles.
There are also equations for the moments of inertia
of rigid objects with shape, and if you need one
of those equations, it will be given to you.
For example, the moment of inertia of a solid sphere
around its center of mass is 2/5 times the mass
of the sphere times the radius of the sphere squared.
Billy, what are the dimensions for moment of inertia?
Billy: Moment of inertia is the sum of the mass
of the particles times the distance
from the axis of rotation squared.
So, kilograms times meters squared.
mr.p: Yes. The dimensions for moment of inertia
are kilograms times meters squared.
Next, Bobby, could you please tell me about
rotational kinetic energy?
Bobby: Sure. Translational kinetic energy equals
1/2 times mass times velocity squared.
So, rotational kinetic energy equals
1/2 times rotational mass, or moment of inertia
times angular velocity squared,
and rotational kinetic energy is in joules.
mr.p: And, we should also look at the concept
of rolling without slipping.
When an object rolls down an incline without slipping,
it gains not only translational kinetic energy
but also rotational kinetic energy.
And, we can look at it in terms of conservation
of mechanical energy.
In other words, a larger moment of inertia
or rotational mass of the object rolling without slipping
will mean a larger rotational kinetic energy.
So, if the object starts out with all
gravitational potential energy,
it will end with both rotational
and translational kinetic energy.
Therefore, if it has a higher rotational kinetic energy,
it will have less translational kinetic energy.
Bobby: Right. The larger the moment of inertia of an object,
the more time it will take to reach the bottom of the hill,
because it will have gained more rotational kinetic energy
and less translational kinetic energy.
And therefore, the object with the larger moment of inertia
will have a smaller linear velocity
at the bottom of the hill.
mr.p: Very nice, Bobby.
You also need to know the equation for the velocity
of the center of mass of a rigid object
that is rolling without slipping.
It is very much like the equation for tangential velocity,
which is on your equation sheet.
It is the velocity of the center of mass
of a rigid object that is rolling without slipping
is equal to the radius of that object
times its angular velocity.
Okay, let's go through another rotational form
of a linear concept: angular momentum.
Linear momentum equals mass times linear velocity.
Therefore, angular momentum equals
rotational mass, or moment of inertia,
times angular velocity, and just like linear velocity,
angular velocity is a vector.
And, the dimensions for angular momentum
are kilograms times meters squared times radians per second.
And because radians are dimensionless, they drop out
and we end up with kilograms times
meters squared divided by seconds.
And, now we can talk about angular impulse.
Linear impulse is equal to the change in momentum,
which is also equal to the force of impact
on the object during the collision multiplied
by the change in time during the collision.
Therefore, angular impulse is equal
to the change in angular momentum of the object,
which is equal to the net torque applied to the object
by the force of impact during the collision
multiplied by the change in time.
[Billy] Mr. P?
mr.p: Yes, Billy?
Billy: What is the symbol for angular impulse?
mr.p: Um, I don't actually believe it has one,
so I just use delta L for the change in angular momentum.
Billy, what are the dimensions for angular impulse?
Billy: Angular impulse. So, torque times change in time
so it's Newtons times meters times seconds.
mr.p: Yes. Angular impulse is in terms of Newtons
times meters times seconds, and because angular impulse
is also change in angular momentum and has the same
dimensions as angular momentum,
kilograms times meters squared over seconds.
We have reached the end of my review
of rotational dynamics for AP Physics 1.
My next review lesson is about universal gravitation.
You are more than welcome to enjoy that video.
You can also visit my website flippingphysics.com
where you can find all of my AP Physics 1
review videos organized with lecture notes.
Thank you very much for learning with me today.
I enjoyed learning with you.