Moments, Levers, Gears and Momentum Changes
Moments and Turning Forces
A moment is the turning effect of a force around a fixed point called the pivot. The size of the moment depends on the force applied and the perpendicular distance from the pivot to the line of action of the force:
$$M = F \times d$$
where \(M\) is the moment in newton metres (Nm), \(F\) is the force in newtons (N) and \(d\) is the perpendicular distance from the pivot in metres (m).
The distance must be the perpendicular distance from the pivot to the line of action of the force — not the distance along the object itself if the force is applied at an angle.
If an object is balanced (in equilibrium), the total clockwise moment about any pivot equals the total anticlockwise moment. This is called the principle of moments.
A child weighing 400 N sits 2.0 m from the pivot of a see-saw. Where must a child weighing 500 N sit to balance it?
Clockwise moment = Anticlockwise moment
$$500 \times d = 400 \times 2.0$$
$$d = \frac{800}{500} = 1.6 \text{ m from the pivot}$$
Levers and Gears
A lever is a simple machine that uses moments to multiply forces. By applying a small force at a large distance from the pivot, a larger force can be produced at a shorter distance from the pivot on the other side. Examples include crowbars, wheelbarrows and scissors.
The longer the lever arm (distance from effort to pivot), the less force is needed to produce the same turning moment. This is why a long spanner makes it easier to undo a tight bolt — the larger distance means a smaller force produces the same moment.
Gears are toothed wheels that transmit rotational motion and force. When two gears are meshed together, they turn in opposite directions. A large gear driving a small gear increases the speed of rotation but reduces the force. A small gear driving a large gear decreases the speed but increases the force (the turning moment).
Gears are force multipliers: a small gear driving a large gear increases the moment because the larger gear has a greater radius, so the force acts at a larger distance from the axle.
Momentum
Momentum is a property of all moving objects. It is calculated using:
$$p = m \times v$$
where \(p\) is momentum in kilogram metres per second (kg m/s), \(m\) is mass in kilograms (kg) and \(v\) is velocity in metres per second (m/s). Momentum is a vector quantity — it has both magnitude and direction.
In a closed system (where no external forces act), the total momentum before an event equals the total momentum after. This is the conservation of momentum. It applies to all collisions and explosions.
In a collision between two objects: \(m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'\). In an explosion where objects start at rest, the total momentum is zero before and after — so the objects move in opposite directions with equal and opposite momenta.
Force, Momentum and Safety
The force acting on an object is equal to the rate of change of momentum:
$$F = \frac{m \Delta v}{\Delta t}$$
This means that to reduce the force during a collision, you can increase the time over which the momentum changes. Car safety features exploit this principle: crumple zones deform slowly, increasing the collision time and reducing the force on passengers. Seat belts stretch slightly, air bags inflate to cushion impact, and cycle helmets contain foam that compresses on impact — all increasing the time of deceleration.
A 1200 kg car travelling at 15 m/s collides with a stationary 800 kg car. After the collision, they move off together. Calculate their shared velocity.
Total momentum before = \(1200 \times 15 + 800 \times 0 = 18{,}000\) kg m/s
Total momentum after = \((1200 + 800) \times v = 2000v\)
$$v = \frac{18{,}000}{2000} = 9 \text{ m/s}$$