Forces and Their Effects: Moments, Levers and Gears
Scalars and Vectors
Physical quantities are classified as either scalar or vector. Scalars have magnitude only; vectors have both magnitude and direction. This distinction matters whenever you add or resolve forces — you must account for direction, not just size.
Scalar examples: speed, mass, temperature, distance, energy. Vector examples: force, velocity, acceleration, displacement, momentum. Force is always a vector — direction is essential when drawing and analysing forces.
Free-Body Diagrams
A free-body diagram shows all forces acting on an object as labelled arrows. Each arrow starts at the object (or its centre of mass) and points in the direction of that force. Arrow length is drawn proportional to force magnitude. Common forces include weight (downward), normal contact force (perpendicular to surface), friction (opposing motion), and tension (along a string or rope).
If the arrows on a free-body diagram all balance — equal and opposite in every direction — the object is in equilibrium. This means it is either stationary or moving at constant velocity.
Step 1: Identify the object in isolation. Step 2: Draw all forces as arrows acting on that single object — not forces the object exerts on other things. Step 3: Label each force with its name and magnitude if known. Step 4: Check for balance: if the vector sum of all forces is zero, the object is in equilibrium; if not, calculate the resultant force using scale or trigonometry.
A Newton-meter (spring balance) is the instrument used to measure force. It works by measuring the extension of a spring and converting this to a force reading using Hooke's law.
Turning Effect and Moments
A force can cause an object to rotate about a fixed point called the pivot (or fulcrum). The turning effect of a force is called its moment:
$$M = F \times d$$
Where \(M\) is the moment in newton-metres (N m), \(F\) is the force in newtons, and \(d\) is the perpendicular distance from the pivot to the line of action of the force in metres. Perpendicular is critical — if the force is not applied at 90° to the lever arm, only the perpendicular component creates a moment.
The principle of moments states that for an object in rotational equilibrium, total clockwise moments = total anticlockwise moments about any pivot.
A child of weight 300 N sits 2 m from the pivot of a see-saw. Where must a second child of weight 400 N sit on the opposite side to balance it?
Clockwise moment = anticlockwise moment
\(400 \times d = 300 \times 2\)
\(400d = 600\)
\(d = 1.5\) m
The 400 N child must sit 1.5 m from the pivot on the opposite side. A heavier person sits closer to the pivot to exert the same moment as a lighter person further away.
Centre of Mass
The centre of mass is the point through which an object's weight effectively acts. For a uniform, symmetrical object — such as a ruler — the centre of mass is at the geometric centre. For irregular objects, it can be found by suspending the object from different points and drawing the plumb-line; the centre of mass is where the lines intersect.
An object will topple if its centre of mass moves outside its base. Wide, low objects (a racing car) are more stable; tall, narrow objects (a double-decker bus on a slope) are less stable. Lowering the centre of mass or widening the base increases stability.
Levers
A lever is a rigid bar that can rotate about a pivot. It uses the principle of moments to provide a mechanical advantage — a small effort force applied at a large distance from the pivot can balance (or exceed) a large load force applied closer to the pivot.
Real-world levers include: a crowbar (pivot at one end, effort at the other end, load near the pivot), a wheelbarrow (pivot at the wheel, load in the bucket, effort at the handles), and a pair of scissors (the pivot is the central bolt; the effort is the handle, the load is the material being cut).
Gears
Gears are toothed wheels that mesh and rotate together. They transmit a turning force (torque) from one axle to another. The key rule is: when a smaller gear drives a larger gear, the larger gear turns more slowly but with a greater turning force — a trade-off between speed and force that mirrors the principle of moments.
The speed ratio between two meshed gears equals the inverse ratio of their numbers of teeth:
$$\frac{\text{speed of driven gear}}{\text{speed of driving gear}} = \frac{\text{teeth on driving gear}}{\text{teeth on driven gear}}$$
A bicycle uses this principle: changing to a larger sprocket (more teeth) on the rear wheel reduces speed but increases the force applied — useful when climbing hills. The pedal gear (chainring) drives the rear sprocket. Adjacent meshed gears always rotate in opposite directions.
A driving gear has 20 teeth and rotates at 300 rpm. It meshes with a driven gear that has 60 teeth. What is the speed of the driven gear?
Speed of driven gear = (teeth on driver ÷ teeth on driven) × speed of driver
= (20 ÷ 60) × 300
= 100 rpm
The driven gear is larger and therefore rotates more slowly — but it applies three times the torque of the driving gear. The two gears also rotate in opposite directions.