Key Equations
Materials - OCR A-Level Physics
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Hooke's law
$$F = kx$$
- Where:
- $F$ = N
- $k$ = N \(m^{-1}\)
- $x$ = m
Only valid up to the limit of proportionality. k is the spring constant (stiffness).
Elastic potential energy (spring constant form)
$$E = \frac{1}{2}kx^{2}$$
- Where:
- $E$ = J
- $k$ = N \(m^{-1}\)
- $x$ = m
Use when spring constant and extension are known. Only valid in the Hooke's law region.
Elastic potential energy (force form)
$$E = \frac{1}{2}Fx$$
- Where:
- $E$ = J
- $F$ = N
- $x$ = m
Equivalent to 1/2 kx^2 when F = kx. Area of the triangle under a linear force-extension graph.
Stress
$$\sigma = \frac{F}{A}$$
- Where:
- $\sigma$ = Pa (N \(m^{-2}\))
- $F$ = N
- $A$ = \(m^{2}\)
A is the cross-sectional area perpendicular to the force. For a circular wire, A = pi(d/2)^2.
Strain
$$\varepsilon = \frac{\Delta L}{L}$$
- Where:
- $\varepsilon$ = dimensionless
- $\Delta L$ = m
- $L$ = m
Strain has no units because it is a ratio of two lengths.
Young modulus
$$\begin{aligned}
E &= \frac{\sigma}{\varepsilon} \\
&= \frac{FL}{A\Delta L}
\end{aligned}$$
- Where:
- $E$ = Pa
A material property (independent of specimen dimensions). Gradient of the linear region of a stress-strain graph.
Springs in series
$$\frac{1}{k_{\text{total}}} = \frac{1}{k_1} + \frac{1}{k_2}$$
- Where:
- $k$ = N \(m^{-1}\)
Springs in series are less stiff (opposite to resistors in series). For n identical springs: k_total = k/n.
Springs in parallel
$$k_{\text{total}} = k_1 + k_2$$
- Where:
- $k$ = N \(m^{-1}\)
Springs in parallel are stiffer (opposite to resistors in parallel). For n identical springs: k_total = nk.