Download Torque Torque is defined as the measure of tendency of a force to

Torque
Torque is defined as the measure of tendency of a force to cause rotational motion.
Torque is the result of a force acting a distance from an axis of rotation and is calculated as
units: Nm
T = Fd⊥
Where F = magnitude of force (N)
d⊥= moment arm (m)
The moment arm is a measure of the perpendicular distance between the line of application of
the force and axis of rotation. This happens to be the shortest distance between the line of
application of the force and axis of rotation.
This is something you already knew! Consider taking off the lug nuts off a wheel:
B
A
At which point, A or B, would you push? Clearly that would be point A. Why? Because you
know that you can exert a greater torque on the lug nut, and hopefully you’ll be able to get the
nut off! If not, what can you do? Consider that you have two options: 1) increase magnitude of
force 2) increase the moment arm. See if you can come up with a solution for how to increase
force and moment arm.
When discussing torque, often we look at how the force is applied about an axis relative to a
resistive force. We then classify different lever systems as being either 1st, 2nd or 3rd class levers.
1st class
2nd class
3rd class
Consider that the thin arrow
represents a resistive force while the thick arrow
motive force. Which of these levers would you use to:
1. Lift a heavy object?
2. Move the end point of the lever at high speeds?
represents a
The body is put together primarily with 3rd class levers. Consider the elbow joint:
Identify the fulcrum, motive & resistive forces, as well as the moment arm for each force?
Can you draw the elbow angle along with associated lever parameters with the elbow in full
extension? At different elbow angles?
What happens to the moment arm of the muscle force when the elbow is fully extended
compared to when the elbow is at 90 degrees?
Does the elbow angle have anything to do with how much weight can be lifted? Why or why
not?
Angular Laws of Motion
Rewriting Newton’s Laws of Motion using angular terminology, we arrive at:
1. Law of Uniform Angular Motion
An object that is in angular motion will continue in its state of angular motion
unless a Torque is applied to it.
2. Law of Angular Acceleration
The angular acceleration an object undergoes is proportional to the Torque
applied to it and inversely proportional to the moment of inertia.
3. Law of Action-Reaction
For every Torque applied, there is an equal and opposite Torque.
Law of Uniform Angular Motion
Consider that you are at the batting cage and about to step into the cage to hit a ball. When
selecting a bat, what do you consider? The weight of the bat? Length? Looks? Considering the
first two parameters, what do they have to do with swinging the bat? Why do we tell kids to
‘choke up’ on the bat? Part of the answer to each of these questions has to do with torque.
Consider the following two ‘bats’:
A.
B.
Now, consider that each bat weighed the same (i.e. they have the same mass). Which would be
harder to swing to hit a ball? Hopefully you said bat B!
If both bats had the same mass, which is the measure of resistance to change motion, why is one
harder to swing than the other? In the angular world, the magnitude of resistance to rotational
motion is not determined only by the mass of the object. Another important descriptor of an
object’s resistance to change rotational motion is how far the mass is from the axis of rotation.
The further from the axis of rotation a mass has, the greater the resistance to change angular
motion. This quality of the object is called “Moment of Inertia.”
Moment of inertia: The measure of an object’s resistance to change angular motion.
To calculate Moment of inertia, you have to consider that an object is made up of a lot of smaller
masses. For example, the bat could be redrawn as:
Axis
Moment of inertia is calculated by summing the product of each mass and the distance from the
axis of rotation squared. In math notation, Moment of Inertia (I) is calculated as:
I = ∑mr2
Units: kgm2
Consider that when swinging the bat to hit a ball, the axis is near the handle of the bat. Clearly
bat B has more mass further away from the axis – which gives it a greater resistance to change
angular motion.
Now, consider that instead of swinging the bats, which would be harder to roll? That is, which
bat, A or B, has a greater moment of inertia if the bats were rolling on a surface? In other words,
which object, A or B, has a greater resistance to change angular motion if the bats were rolled?
The first step in answering this question is to identify the location of the axis of rotation. Next,
conceptualize how the mass of the bat is distributed about the axis. Can you identify which bat
would be easier to roll? Is the moment of inertia the same for each bat when swinging and
rolling it?
Law of Angular Acceleration
When a torque is applied to an object, the change in angular motion is proportional to the torque
and inversely proportional to the moment of inertia. Or, in mathematical terms:
∑T = Iα
I = moment of inertia (kgm2)
α = angular acceleration (radians/s2)
Therefore, units are: kgm2/s2 = m(kgm/s2) = Nm
Rewriting angular acceleration as the rate of change of angular velocity, we arrive at:
∑T = ∆(Iω)/∆t
Where Iω = angular momentum
Angular momentum is defined as the quantity of angular motion. It is calculated as:
L = Iω
units: kgm2/s
Where I = moment of inertia (kgm2)
ω = angular velocity (radians/s)
The angular impulse-momentum relationship is therefore:
∑T∆t = ∆(Iω)
Consider the flight phase of a springboard dive. Given that the diver rotates about the COM,
what external torques are applied to the diver? Considering the only external force acting on the
diver is gravity, and it acts through the COM (causing no torque), there are NO external torques
applied to the diver. Therefore, T∆t = 0 Nm/s
The significance of this is that the change in angular momentum during the flight phase of diving
(or any sport that has a flight phase) has to be zero. But wait a second, doesn’t a diver change
how fast he or she is spinning during a dive? Yes, divers do change angular velocity. Can you
explain how it is that a diver can change angular velocity with no external torques?