Whenever you push a door open, use a spanner to loosen a nut, or balance on a seesaw, you are using the turning effect of a force. In physics, we call this turning effect a moment.
What you'll learn:
- How to define and calculate the moment of a single force.
- Where to find the centre of mass for uniform objects.
- How to use the Principle of Moments to solve balancing problems.
- What a "couple" is and how to calculate its moment.
The Moment of a Force
A force doesn't just push or pull things in a straight line; if it acts at a distance from a fixed pivot, it causes the object to rotate. The strength of this turning effect depends on two things: how big the force is, and how far away it is applied from the pivot.
Moment of a force
The moment of a force about a point is defined as the force multiplied by the perpendicular distance from the point to the line of action of the force.
In mathematical terms, we write this as:
Moment=F×d \text{Moment} = F \times d Moment=F×dWhere:
- FFF is the force in newtons (N).
- ddd is the perpendicular distance in metres (m).
- The unit of a moment is the newton-metre (N m).
The line of action is an imaginary line extending infinitely along the direction the force vector is pointing. To find the correct distance ddd, you must draw a line from the pivot that meets this line of action at a perfect right angle (90∘90^{\circ}90∘).
Using the wrong distance
If a force is applied at an angle (not perfectly perpendicular to the object), you cannot simply multiply the force by the length of the object. You must either use trigonometry to find the perpendicular distance ddd, or resolve the force into a component that is perpendicular to the object.
Calculating a moment at an angle
A student pushes a door handle with a force of 40 N. The handle is 0.80 m from the door's hinges (the pivot). The student pushes at an angle of 60∘60^{\circ}60∘ to the surface of the door. Calculate the moment of the force about the hinges.
- Identify the variables: The force F=40 NF = 40 \text{ N}F=40 N, the distance along the door is 0.80 m, and the angle is 60∘60^{\circ}60∘.
- Find the perpendicular distance (ddd): Imagine the line of action of the force. The perpendicular distance from the hinges to this line forms a right-angled triangle.
- Calculate the moment: Multiply the force by this perpendicular distance.
- State the final answer: Rounding to two significant figures, the moment is 28 N m.
Centre of Mass
Before we look at objects balancing, we need to consider the weight of the objects themselves. A heavy wooden beam has mass distributed all the way along its length. To make calculations simpler, we imagine all of this mass is concentrated at one specific point.
Centre of mass
The centre of mass is the single point through which the entire weight of an object can be considered to act.
For A-Level Physics, you are often asked about a uniform regular solid (like a perfectly straight, perfectly even metal bar). For a uniform regular solid, the centre of mass is exactly at its geometric centre. For example, if you have a uniform beam of length 6.0 m, you can draw a single downward force arrow for its weight exactly at the 3.0 m mark.
The Principle of Moments
When an object is in equilibrium (it is balanced and not rotating), the forces trying to twist it clockwise are perfectly canceled out by the forces trying to twist it anticlockwise.
The Principle of Moments
For an object in equilibrium, the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about that same point.
You can choose any point on the object to act as your pivot for these calculations. A clever trick is to choose a point where an unknown force acts; because the distance from that point to itself is zero, the moment of that unknown force becomes zero, eliminating it from your equation!
Balancing a uniform beam
A uniform beam of length 5.0 m and mass 20 kg is supported horizontally by two pillars, one at the far left end (A) and one placed 1.0 m from the right end (B). Calculate the upward reaction force from pillar B.
- Calculate the weight of the beam: Convert the mass to a weight using g=9.81 m s−2g = 9.81 \text{ m s}^{-2}g=9.81 m s−2.
- Locate the centre of mass: Because the beam is uniform, its weight acts perfectly in the middle, which is 2.5 m from the left end (A).
- Choose a pivot: Take moments about point A. This is useful because the unknown reaction force at A has a distance of 0 m from A, so its moment is zero.
- Set up the moments equation: Identify clockwise and anticlockwise moments about A.
- Clockwise moment comes from the weight of the beam: 196.2 N×2.5 m196.2 \text{ N} \times 2.5 \text{ m}196.2 N×2.5 m.
- Anticlockwise moment comes from the upward push of pillar B (RBR_BRB). Pillar B is 4.0 m away from A (since it is 1.0 m from the right end of a 5.0 m beam).
- Apply the Principle of Moments:
- Solve for RBR_BRB:
Rounding to two significant figures, the reaction force at B is 120 N.
Couples
Sometimes you want to rotate an object without pushing it sideways. Think about turning a steering wheel: your left hand pushes up while your right hand pulls down. You are applying a pair of forces.
Couple
A couple is a pair of equal and opposite coplanar forces.
Because the two forces are perfectly equal and point in exactly opposite directions, their combined linear push is zero. The object won't accelerate off in a straight line; it will strictly rotate.
To find the turning effect of a couple, AQA requires a specific definition:
Moment of a couple = force ×\times× perpendicular distance between the lines of action of the forces.
Couples only rotate
Don't multiply the total force by the distance! The formula only uses one of the forces. If you have a 30 N force pushing up and a 30 N force pulling down, you use F=30 NF = 30 \text{ N}F=30 N in the formula, not 60 N.
Turning a T-bar wrench
A mechanic uses a T-bar wrench to loosen a wheel nut. They apply a force of 150 N forwards with their left hand and 150 N backwards with their right hand. The distance between their hands is 40 cm. Calculate the moment of the couple applied to the wrench.
- Identify the force and distance: The force F=150 NF = 150 \text{ N}F=150 N. The perpendicular distance between the lines of action is s=40 cms = 40 \text{ cm}s=40 cm.
- Convert units to SI standard: Always work in metres.
- Calculate the moment of the couple:
In the exam
- Draw a quick, clear diagram: Even if one is provided, draw forces directly onto it. Mark the centre of mass for any uniform objects with a downward arrow.
- State your pivot: Write "Taking moments about point X..." at the start of your working. This makes your logic clear to the examiner and helps you avoid confusing your own distances.
- Watch out for masses: Look closely at the units in the question. If you are given kilograms (kg), you must multiply by 9.81 to get the weight force in newtons (N) before calculating moments.
Check yourself
- What are the two specific things you must multiply together to calculate the moment of a single force?
- If an object is a "uniform regular solid", exactly where does its weight act?
- When calculating the moment of a couple, do you add the two forces together?
- What does the Principle of Moments state about an object in equilibrium?
