Forces in Action: Springs, Hooke's Law and Moments
Elastic and Plastic Deformation
To stretch, bend or compress an object, more than one force must be applied. A single force would simply cause the object to accelerate (Newton’s Second Law). Two opposing forces are required for deformation — for example, a weight pulling a spring downwards while the support holds it up.
When forces are applied to a spring, it can undergo two types of deformation:
• Elastic deformation — the object returns to its original shape when the force is removed. The spring ‘bounces back.’
• Plastic deformation — the object is permanently deformed. The spring has been stretched beyond its elastic limit and does not recover.
Hooke’s Law
Hooke’s Law states that the extension of a spring is directly proportional to the force applied, as long as the elastic limit is not exceeded. The equation is:
$$F = k \times e$$
Where \(F\) is force (N), \(k\) is the spring constant (N/m), and \(e\) is extension (m). Extension is the change in length: \(e = \text{stretched length} - \text{natural length}\). A larger spring constant means a stiffer spring. On a force-extension graph, the straight-line region confirms Hooke’s Law; the gradient of this line equals the spring constant \(k\). Beyond the elastic limit, the graph curves — deformation becomes plastic.
Hooke’s Law: \(F = ke\). The gradient of the straight-line section of a force-extension graph equals the spring constant \(k\). Beyond the elastic limit, deformation is plastic and permanent.
Clamp a spring vertically from a stand. Measure and record the natural length using a ruler. Add known masses one at a time, recording the total length each time. Calculate extension for each mass (\(e = \text{new length} - \text{natural length}\)). Plot force (y-axis) against extension (x-axis). The straight-line section confirms Hooke’s Law; the point where the line begins to curve marks the elastic limit. To improve accuracy, take repeated readings at each force and calculate the mean; read the ruler at eye level to avoid parallax; ensure the spring hangs freely without touching the stand.
When a spring is stretched elastically, energy is stored as elastic potential energy (EPE):
$$E_e = \frac{1}{2} \times k \times e^2$$
Where \(E_e\) is elastic potential energy (J), \(k\) is the spring constant (N/m), and \(e\) is extension (m). This equation only applies within the elastic limit (the linear region). On a force-extension graph, the EPE equals the triangular area under the straight-line section: \(\frac{1}{2} \times F \times e\). When the spring is released, this stored energy is transferred to kinetic energy or gravitational potential energy.
Moments
A moment is the turning effect of a force about a pivot. It is calculated using:
$$M = F \times d$$
Where \(M\) is the moment (N m), \(F\) is the force (N), and \(d\) is the perpendicular distance from the pivot to the line of action of the force (m). The distance must be perpendicular to the force — using a slanted distance gives the wrong answer. Moments are measured in newton-metres (N m).
For an object to be in rotational equilibrium, the total clockwise moment about any pivot must equal the total anticlockwise moment about the same pivot. This is the principle of moments:
$$\text{total clockwise moment} = \text{total anticlockwise moment}$$
Question: A see-saw has a pivot at its centre. A child of weight 300 N sits 1.5 m from the pivot on the left. How far from the pivot must a 450 N adult sit on the right to balance the see-saw?
Step 1: Anticlockwise moment (child) = \(300 \times 1.5 = 450 \text{ N m}\).
Step 2: For equilibrium: clockwise moment = anticlockwise moment.
\(450 \times d = 450\)
\(d = 450 \div 450 = 1.0 \text{ m}\).
The adult must sit 1.0 m from the pivot. Notice that the heavier person sits closer to the pivot to produce the same moment.
Moment (N m) = Force (N) × perpendicular distance from pivot (m). For equilibrium: total clockwise moment = total anticlockwise moment (principle of moments). Distance must be perpendicular to the force.
Levers and Gears
A lever uses the principle of moments to act as a force multiplier. By applying a small effort at a large distance from the pivot, you can overcome a larger load closer to the pivot. For example, a crowbar enables a small downward push to lift a large rock. Scissors, tweezers, a bottle opener and a wheelbarrow are all examples of levers in different configurations.
The key idea: the lever multiplies force because moment = force × distance. If the effort arm is longer than the load arm, the effort force needed is smaller than the load force, even though both produce equal moments.
Gears are toothed wheels that mesh together and transmit turning forces. When two gears of different sizes mesh:
• The smaller gear turns faster but with less force (smaller moment).
• The larger gear turns more slowly but with greater force (larger moment).
• The number of teeth on each gear determines the gear ratio — larger gear ÷ smaller gear. A higher gear ratio means a greater increase in force but a reduction in speed.