You can begin to see how, in a sense, we have solved all simple-harmonic-motion problems once we have solved the problem of the horizontal spring. The restoring force of a spring, $F_{\text{Sp}}=-kx$, is directly proportional to the displacement x from equilibrium. The pendulum’s restoring force, in the small-angle approximation, is directly proportional to the displacement s. A restoring force that is directly proportional to the displacement from equilibrium is called a linear restoring force. For any linear restoring force, the equation of motion is identical to the spring equation (other than perhaps using different symbols). Consequently, any system with a linear restoring force will undergo simple harmonic motion around the equilibrium position.

This is why an oscillating spring is the prototype of SHM. Everything that we learn about an oscillating spring can be applied to the oscillations of any other linear restoring force, ranging from the vibration of airplane wings to the motion of electrons in electric circuits.

MODEL 15.1 Simple harmonic motion

For any system with a restoring force that’s linear or can be well approximated as linear.

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• Motion is SHM around the equilibrium position.

• Frequency and period are independent of the amplitude.

• Mathematically:

  • For an appropriate position variable u, the equation of motion can be written

    $$d^{2}u/d{t}^2=-Cu$$

    where C is a collection of constants.

  • The angular frequency is $\omega =\sqrt{C}$.

  • The position and velocity are

    $$u=A\text{cos}(\omega t+{\varphi }_0)v_u=-v_{\text{max}}\text{ sin}(\omega t+{\varphi }_0)$$

    where A and ${\varphi }_0$ are determined by the initial conditions.

  • Mechanical energy is conserved.

• Limitations: Model fails if the restoring force deviates significantly from linear.

  

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Workbook Exercise 22