Today we’ll be talking about:
Let’s strap in!
Previously, we’ve seen forces acting on objects, causing them to MOVE (translate).
Now, we’ll see that forces can also cause objects to SPIN (rotate).
What is Moment?
Product of a force & the perpendicular distance from a pivot to the line of action of the force.
It is the TURNING EFFECT of a force.
Moment = F x d
Sometimes a force’s line of action is NOT perpendicular to the lever attached to the pivot.
There are 2 ways you can find the moment:
|Finding the perpendicular distance between the line of action & the pivot.||The same as the previous figure, but with a dotted line connecting the arrow of the force to the pivot point. This line is at a right angle to the force and has a length labelled as d sin θ.|
The dotted line is at a right angle to the line of action of force F.
As you can see, this distance can be found using trigonometry.
dperp = d sin θ
moment = F x d sin θ
|Finding the component of the force which is perpendicular to the lever.|
You can find this component using trigonometry.
Depending on what you’re given, you can use:
Fperp = F cos θ1
Fperp = F cos (90 – θ)
moment = F cos θ1 x d
Make sure you identify the correct angle θ before using these equations.
The angle given may be at a different point, perhaps at the pivot instead of at the force.
Don’t relay on these equations; instead use your knowledge of trigonometry to solve them!
Direction of moment
Moment can be said to be CLOCKWISE or ANTICLOCKWISE.
What is Torque?
The total turning effect of all forces on an object.
On a lever/beam, there may be multiple forces acting on different sides.
You can find the torque (NET moment) by:
- Identifying the total clockwise moment
- Identifying the total anticlockwise moment
- Finding the difference & direction of resultant moment
What is a Couple?
A pair of forces that are equal in magnitude that act exactly in the opposite direction.
The pair cannot act on the same POINT (because they would cancel out), but on the same OBJECT at different points.
It causes ROTATIONAL MOTION.
2 fingers turning a screwdriver.
One finger applies a force in one direction, while the other applies the same force in the opposite direction.
If only one force was applied, the screwdriver may move (translate) instead of rotate in place.
What is Torque of a Couple?
The net turning effect of a COUPLE.
Both forces in a couple have a moment about the pivot.
Torque is the sum of both of these moments.
Torque = moment1 + moment2
Let’s call the distance between each force & the pivot “r” for radius.
Torque = Fr + Fr (since the forces are equal)
Torque = 2Fr
Since 2r = perp. distance between the 2 forces,
Torque = F x perp. distance between forces
Note that here, you don’t have to multiply the force by 2, just choose 1 force from the couple.
It doesn’t matter if the position of the pivot is not equidistant from both forces,
no matter where the pivot is, the magnitude of the torque is the same.
(as long as the pivot is anywhere BETWEEN the two forces).
Sometimes the forces are NOT PERPENDICULAR to the beam.
Just like with moments, there are 2 ways to find the torque of a couple:
|Finding the PERPENDICULAR DISTANCE between both forces.
dperp = 2r cos θ
|Finding the PERPENDICULAR COMPONENT of both forces (which would be equal).
Fperp = F sin θ
Remember that a couple MUST have 2 forces in EXACTLY opposite directions, so both angles θ1 & θ2 would be the same.
What is the Principle of Moments?
“On a balanced system, the algebraic sum of the moments of all forces acting on the system at any point is 0.”
Thus, total clockwise moment = total anticlockwise moment.
A system in rotational equilibrium does NOT have any rotation,
Thus, torque = 0.
What is the Centre of Gravity?
A single point on a body where the weight acts // seems to be concentrated.
If you were to balance an object on this point, it would be in rotational equilibrium (does not rotate) (torque = 0).
This is because the weight on either side of this point are equal.
Since the weights of different parts of the object act as forces about the centre of gravity, these weights cause moments in each part of the object.
If the object is in rotational equilibrium, the net moment = 0.
Thus, the weights must be equal.
Now that we’ve seen rotational equilibrium in action (or rather, inaction), let’s check out translational equilibrium.
When is an object in translational equilibrium?
When the sum of forces in any direction is 0.
No acceleration occurs.
3 forces in equilibrium would form a complete, continuous triangle if joined tip-to-tail.
The 3rd force effectively cancels out the resultant of the first 2 forces.
You can also trace the vectors from the origin, & follow them along the arrows to return to the origin.
The net vector is 0.
When is an object in COMPLETE Equilibrium?
When ALL aspects of its motion are balanced.
An object can have 2 types of motion:
- Translational Motion
- Rotational Motion
For an object to be in complete equilibrium, BOTH of these aspects must be in equilibrium.
- The sum of forces in any direction must be 0 (translational equilibrium)
- The sum of moments about any point must be 0 (rotational equilibrium)