Explain the Hook's law in Alevel further mathematics mechanics
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Hooke's Law is a fundamental principle in mechanics that describes how elastic materials respond to applied forces. When we apply a force to a spring or elastic string, it extends proportionally to that force, provided we don't exceed the material's elastic limit. This relationship is crucial for understanding mechanical systems in A-Level Further Mathematics.
Hooke's Law can be expressed in two mathematical forms. The first form, F equals k times x, uses the spring constant k, which is specific to each spring or elastic material. The second form, F equals lambda x over L, uses the modulus of elasticity lambda and natural length L. These forms are equivalent since k equals lambda over L. The graph shows the linear relationship between force and extension, where the slope represents the spring constant.
Now let's observe how a spring behaves under increasing force. As we gradually increase the applied force, the spring extends proportionally. Notice that the extension increases linearly with the force - this demonstrates Hooke's Law in action. The natural length is the spring's length with no applied force, and any extension beyond this follows the linear relationship F equals k times x.
It's crucial to understand that Hooke's Law only applies within the elastic limit of a material. Beyond this point, the material undergoes plastic deformation and won't return to its original shape. The graph shows the linear Hooke's Law region in blue, followed by the non-linear plastic deformation in red. In A-Level Further Mathematics, Hooke's Law is used in equilibrium problems, dynamics, oscillations, and energy calculations, particularly for elastic potential energy given by half k x squared.
Let's work through a typical A-Level problem. We have a spring with modulus of elasticity 20 Newtons and natural length 0.5 meters, extended by 0.2 meters. First, we find the spring constant using k equals lambda over L, giving us 40 Newtons per meter. Next, we calculate the tension using F equals k times x, which gives us 8 Newtons. Finally, the elastic potential energy is half k x squared, equal to 0.8 Joules. This demonstrates the practical application of Hooke's Law in solving mechanics problems.