Force Law, Springs and Energy in SHM: Quick Revision

A brief note on Force Law, Springs and Energy in SHM. Read it before a test. Try each question at the end before you open its answer.

The force law

• Simple harmonic motion needs a restoring force in proportion to the displacement: $F = -kx$.
• Angular frequency: $\omega = \sqrt{k/m}$.
• Period: $T = 2\pi\sqrt{m/k}$, and $f = 1/T$.
• The period does not depend on the amplitude, or on $g$.

Springs

• Hanging spring: at rest $kd = mg$, and $T = 2\pi\sqrt{d/g}$.
• Side by side, in parallel: $k = k_1 + k_2$.
• End to end, in series: $\dfrac{1}{k} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$.
• A block between two springs and two walls: parallel.
• A piece of a spring is stiffer: $k \times$ length stays the same.
• Two bodies on one spring: use $\mu = \dfrac{m_1m_2}{m_1 + m_2}$.

Energy

• Potential energy: $U = \tfrac12 kx^2$.
• Total energy: $E = \tfrac12 kA^2$.
• Speed: $v = \omega\sqrt{A^2 - x^2}$, largest $v_{max} = \omega A$ at the middle.
• $K = U$ at $x = A/\sqrt2$.
• Each energy repeats every $T/2$ and averages to $E/2$.

A worked example

A block of $0.30\ \text{kg}$ on a spring of $120\ \text{N/m}$ swings with an amplitude of $4.0\ \text{cm}$.

1. Angular frequency. $\omega = \sqrt{\dfrac{120\ \text{N/m}}{0.30\ \text{kg}}} = 20\ \text{rad/s}$.
2. Period. $T = \dfrac{2\pi}{20\ \text{rad/s}} = 0.314\ \text{s}$.
3. Energy. $E = \tfrac12 \times 120\ \text{N/m} \times (0.040\ \text{m})^2 = 0.096\ \text{J}$.
4. Largest speed. $v_{max} = 20\ \text{rad/s} \times 0.040\ \text{m} = 0.80\ \text{m/s}$.

Practice with new numbers

A program to try

The program works through the example above.

Mistakes to avoid

• Writing $T = 2\pi\sqrt{k/m}$. The mass goes on top.
• Adding spring constants in series. In series, add the inverses.
• Forgetting to square the amplitude in $\tfrac12 kA^2$.
• Putting equal energies at $A/2$. They are equal at $A/\sqrt2$.

Check yourself

1. A block on a spring has a period of $0.5\ \text{s}$. The spring is swapped for one four times as stiff. What is the new period?

2. Two springs of $30\ \text{N/m}$ and $60\ \text{N/m}$ are joined end to end. What is the spring constant of the pair?

3. The amplitude of an oscillator is halved. What happens to its largest speed and to its energy?

4. Why does a hanging spring have the same period as the same spring on a smooth floor?