Mr.p: Good morning, let's learn about torque.
"Flipping Physics" Intro
But before we do, we need to review translational and rotational motion.
Bobby, what is translational motion?
Bobby: Translational motion is like when something moves from one place to another, right?
Bo: I think we need to refer to the center of mass of the object.
Bobby: Yeah, so translational motion is when the center of mass of an object moves from one place to another.
Billy: And rotational motion is when the object moves in circles around its center of mass.
Bobby: An object could have both translational and rotational motion, right?
Mr.p: You are correct. Let's show some examples.
This bicycle wheel is in rotational motion, but not translational motion.
This bicycle wheel is in translational motion because its center of mass is in motion.
This bicycle wheel is in both translational and rotational motion.
As you recall, force is the ability to cause acceleration.
Well, torque is essentially rotational force, it is the ability of a force to cause angular acceleration.
Watch as I apply a torque to a bicycle wheel, twice.
Billy, please explain how it is that you know I applied a net torque to the bicycle wheel, twice.
Billy: First the wheel was at rest, then you started the wheel spinning,
so you changed the angular velocity of the wheel and that means you angularly accelerated the wheel,
which means you applied a net torque to the wheel to cause that angular acceleration.
And then you did the reverse. The wheel was spinning and you stopped it.
So again, you applied a net torque to the wheel to angularly accelerate the wheel;
you changed the angular velocity of the wheel to zero.
Mr.p: Billy, that's very nice.
The symbol for torque is the lowercase Greek letter tau. It looks like a curly t. And torque equals r F sine theta.
F is the force causing the torque.
This r perpendicular is called either the moment arm or the lever arm and it equals "r" times sine theta.
This means that sometimes the equation for torque is given as the moment arm times the force.
The r vector is the position vector from the axis of rotation to the point the force is applied to the object.
And theta is the angle between the direction of the r vector and the direction of the force vector.
And the arrow symbol over tau means torque is a vector, which means class torque has both...
Billy and Bo: Magnitude and Direction.
Bobby: That's great. But, I have no idea what you are talking about with that r vector moment arm lever thing.
Mr.p: Right. There is a lot here.
So let's look at some specific examples to help us understand all of this.
I like to say everything you ever needed to know about torque,
you already know because, you have opened many doors.
So let's look at a door.
Through trial and error you have honed your ability to walk through a door.
When you approach a door, class, what do you look for?
Billy: The handle.
Mr.p: And the handle is located where relative to the hinge, which is the axis of rotation?
Bo: The handle is far away from the hinge.
Mr.p: Right.
The handle of the door is about as far away from the axis of rotation, or hinge,
as the handle can get on the door.
When you open a door you push near the handle.
You have learned not to try to open the door by pushing near the hinge or axis of rotation.
Let's talk about why that is.
The magnitude of the "r" value is from the axis of rotation to where the force is applied.
So, when we apply a force at the door handle,
there is a larger "r" value, r1,
than if we apply a force near the axis of rotation, which has a smaller "r" value, r2.
If the force applied has the same magnitude and direction,
Billy, how do the torques compare at positions 1 and 2?
Billy: It looks like the angle between the force direction and the r vector for both positions are 90 degrees
and the sine of 90 degrees equals 1. So the torque in this case is just the "r" value times the force applied.
Because the r value for position 2 is so much smaller than the r value for position 1,
the torque when we push on the door closer to the axis of rotation is much smaller.
Mr.p: And what does that mean for being able to open the door?
Billy: A smaller torque means a smaller ability to cause an angular acceleration.
So when we push near the hinge,
the torque has a smaller ability to cause an angular acceleration to open the door.
Mr.p: Absolutely Billy, thank you.
Another thing we have learned to never do to try to open a door is to push at an angle like this.
Bobby, why will we never try to open a door by pushing at an angle like this?
Bobby: Okay, so when we push it an angle, that's not 90 degrees, ...
Oh, because of the shape of the sine curve.
The maximum value for the sine of an angle is at 90 degrees.
So, at an angle that is not 90 degrees the sine of the angle will be less, which will make the torque less,
which means the ability to cause an angler acceleration to open the door is less.
Bo: Why is the angle theta on that side of the force?
I thought he said it was between the force and the r vector.
Bobby: Oh theta is between the direction of the force and the direction of the r vector.
The r vector points to the left, so theta is between to the left and the direction of the force.
Bo: Okay, that makes sense. Thanks.
Bobby: You're welcome.
Mr.p: Very nice Bobby.
You are correct that because the sine of 90 degrees results in the maximum value for the sine of the angle,
an angle of 90 degrees between the direction of the force vector and the direction of the "r" vector
will result in the maximum torque.
Okay, one last example of how you will never try to open the door.
Bo, why will you never try to open a door by pushing on the side of the door like this?
Bo: Well, in that case the angle between the "r" value and the force applied is 180 degrees
and the sine of 180 degrees is zero.
So no matter how hard you push, that will never cause a torque on the door.
Mr.p: Correct, no matter how hard I push or even pull on the side of the door,
the angle will be either 180 degrees for a push or zero degrees for a pull.
Regardless, the sine of zero and 180 degrees are both zero, so there is no torque and
no ability to cause an angular acceleration of the door.
Bo: mr.p?
Mr.p: Yes, Bo?
Bo: What about moment arm or lever arm? You have not mentioned that at all.
Mr.p: Right.
Let's go back to the example with the force at an angle to talk about the moment arm.
The "r" position vector is the hypotenuse of a right triangle, which has two sides;
one side is a line that includes the force and the extension of the force causing the torque.
The other side is the component of the r vector, which is perpendicular to the extension line of the force.
This perpendicular component of r is called the moment arm or the lever arm.
Notice in this example that the moment arm is smaller than the "r" vector
because the angle in our torque equation is not 90 degrees.
Bobby: Oh, and when the angle is 90 degrees, like the first door example,
the "r" value and moment arm are equal in magnitude.
Bobby: And when the angle is 180 degrees, like in the last example where you pushed on the side of the door,
the moment arm is actually zero.
Bo: I agree with all that, but that does not really explain what the moment arm is.
Billy: Yeah. Bobby: Right.
Mr.p: Okay, the moment arm is defined as
the perpendicular distance between the line of action of the force and the axis of rotation.
Bo: Right. Bobby: Sure.
Billy: So, the dotted line extending from the force must be the "line of action" of the force.
And the moment arm is the perpendicular distance from the axis of rotation to the "line of action" of the force.
Perpendicular because it is a right triangle.
Bobby: And sine of theta equals opposite over hypotenuse or moment arm over "r" value.
So the moment arm equals r sine theta.
Bo: I thought theta was on other side.
Bobby: Right.
Billy: Those two angles, let's call them theta one and theta two,
are supplementary angles, the signs of supplementary angles have the same value.
So, it does not matter if you use theta 1 or theta 2.
Bo: What?
Bobby: Yeah, that's right. Look at the shape of the sine curve,
the sine of 90 plus x and the sine of 90 minus x give the same value.
90 plus x and 90 minus x are supplementary angles.
They add up to 180 degrees. Because the signs of supplementary angles have the same value,
It does not matter if you use theta 1 or theta 2 in the equation sine theta equals opposite over hypotenuse.
Bo: Okay.
Mr.p: And lastly, Bo, please determine the units for torque.
Bo: Torque equals moment arm times force, so meters times newtons.
Bobby: Aren't those joules?
Mr.p: Typically we use newton meters, and yes, Bobby, newton meters are joules.
However, because torque is a vector and entirely different from work and energy, which are scalars,
we use newton meters for torque and joules for work and energy.
It is easier to identify torque as different from work and energy when we use different units for each.
Thank you very much for learning with me today, I enjoyed learning with you.