IB MYP 4-5 Physics- Hooke’s law - Study Notes - New Syllabus
IB MYP 4-5 Physics-Hooke’s law – Study Notes
Key Concepts
- Hooke’s law
Hooke’s Law
Hooke’s Law
Hooke’s Law is a fundamental principle in mechanics and materials science that relates the force applied to an elastic object to the amount it deforms. It is especially important for springs, elastic bands, and other materials that return to their original shape after being stretched or compressed, provided they are not overstressed.
- Historical Note: Proposed by Robert Hooke in 1660, published in 1678, expressed in Latin as “ut tensio, sic vis” meaning “as the extension, so the force”.
- Basic Idea: The extension (or compression) of an elastic object is directly proportional to the applied force, as long as the elastic limit is not exceeded.
Mathematical Form
F = k \cdot x
- \(F\) = applied force (N) or restoring force
- \(k\) = spring constant (N/m), a measure of stiffness (larger \(k\) → stiffer spring)
- \(x\) = extension (+) or compression (−) from natural length (m)
Important Related Quantities
- Stress: \( \sigma = \dfrac{F}{A} \) → force per unit cross-sectional area (Pa)
- Strain: \( \varepsilon = \dfrac{\Delta L}{L_0} \) → fractional change in length (no unit)
- In elastic materials, stress is proportional to strain: \( \sigma \propto \varepsilon \) (Young’s modulus law)
Graphical Representation
- Load–Extension Graph: A straight line through the origin in the elastic region.
- Proportional Limit: The maximum point where Hooke’s Law is valid.
- Elastic Limit: Beyond this, the material will not return to original shape.
Elastic Potential Energy in a Spring
The work done to stretch or compress a spring is stored as elastic potential energy:
U = \dfrac{1}{2} k x^2
- This energy is recoverable if deformation remains within elastic limit.
- If deformation exceeds the elastic limit, some energy is lost as heat or permanent damage.
Limitations of Hooke’s Law
- Applies only for small deformations.
- Material must be elastic (returns to original shape).
- Fails beyond elastic limit — permanent deformation occurs.
Applications of Hooke’s Law
- Designing suspension systems in vehicles (shock absorbers).
- Measuring force using spring balances.
- Calibrating measuring instruments.
- Seismographs for detecting earthquakes.
- Determining material stiffness for engineering purposes.
Example:
A spring stretches by \(0.05\ \text{m}\) when a force of \(10\ \text{N}\) is applied. Find the spring constant and plot the load-extension relationship up to \(F = 20\ \text{N}\).
▶️ Answer/Explanation
Using \(F = kx\):
k = \dfrac{F}{x} = \dfrac{10}{0.05} = 200\ \text{N/m}
So, the equation becomes \(F = 200x\).
For \(F = 20\ \text{N}\), extension \(x = \dfrac{20}{200} = 0.1\ \text{m}\).
\(\boxed{k = 200\ \text{N/m}}\)
Example:
A spring of spring constant \(150\ \text{N/m}\) is compressed by \(0.1\ \text{m}\). Find the elastic potential energy stored in it.
▶️ Answer/Explanation
U = \dfrac{1}{2} k x^2
U = \dfrac{1}{2} (150) (0.1)^2
U = \dfrac{1}{2} (150) (0.01) = 0.75\ \text{J}
\(\boxed{U = 0.75\ \text{J}}\)