# 17.7 Energy Conservation in SHM This section explores the principle of energy conservation for a body executing Simple Harmonic Motion (SHM), using a mass-spring system as the primary example. The total mechanical energy of the system remains constant, oscillating between kinetic energy (K.E.) and potential energy (P.E.). ## Potential Energy (P.E.) in SHM The work done to stretch or compress a spring by a displacement $x$ is stored as elastic potential energy. - **Applied Force (Hooke's Law):** $F = kx$ - **Average Force:** Since the force increases linearly from $0$ to $kx$, the average force is $F_{av} = \frac{1}{2} kx$. - **Work Done (Stored P.E.):** $W = F_{av} \cdot x = (\frac{1}{2} kx)(x) = \frac{1}{2} kx^2$ - **Potential Energy Formula:** $$P.E. = \frac{1}{2} k x^2$$ ## Kinetic Energy (K.E.) in SHM The kinetic energy of the mass depends on its instantaneous velocity. - **Instantaneous Velocity:** The velocity $v$ at any displacement $x$ is given by $v = \omega \sqrt{x_o^2 - x^2}$, where $x_o$ is the amplitude and $\omega = \sqrt{k/m}$. - **Kinetic Energy Formula:** $$K.E. = \frac{1}{2} mv^2$$ Substituting the expression for $v$: $$K.E. = \frac{1}{2} m \left( \frac{k}{m}(x_o^2 - x^2) \right) = \frac{1}{2} k (x_o^2 - x^2)$$ ## Total Energy (T.E.) and Conservation The total energy of the system is the sum of its kinetic and potential energies at any instant. - **Total Energy Calculation:** $$T.E. = K.E. + P.E.$$ $$T.E. = \frac{1}{2} k (x_o^2 - x^2) + \frac{1}{2} k x^2$$ $$T.E. = \frac{1}{2} k x_o^2$$ - Since $k = m\omega^2$, an alternative form is: $$T.E. = \frac{1}{2} m \omega^2 x_o^2$$ - **Conservation Principle:** The total energy of a system in SHM is constant and is proportional to the square of the amplitude of oscillation ($x_o^2$). ## Energy at Key Positions | Position | Displacement ($x$) | Velocity ($v$) | Potential Energy (P.E.) | Kinetic Energy (K.E.) | Total Energy (T.E.) | | :--- | :---: | :---: | :--- | :--- | :--- | | **Mean** | $x = 0$ | Max ($v_o$) | $0$ | Max ($\frac{1}{2} k x_o^2$) | $\frac{1}{2} k x_o^2$ | | **Extreme** | $x = \pm x_o$ | $0$ | Max ($\frac{1}{2} k x_o^2$) | $0$ | $\frac{1}{2} k x_o^2$ | | **Any Point** | $x$ | $v$ | $\frac{1}{2} k x^2$ | $\frac{1}{2} k (x_o^2 - x^2)$ | $\frac{1}{2} k x_o^2$ | The energy continuously transforms between kinetic and potential forms, as illustrated by the graph: - In Simple Harmonic Motion, the total mechanical energy is conserved. - Energy continuously oscillates between potential energy (stored in the spring) and kinetic energy (of the mass). - Potential energy is maximum at the extreme positions ($x = \pm x_o$) and zero at the mean position ($x = 0$). - Kinetic energy is maximum at the mean position and zero at the extreme positions. - The total energy is constant at all points and is directly proportional to the square of the amplitude. | Formula | Description | | :--- | :--- | | $P.E. = \frac{1}{2} k x^2$ | Potential Energy at displacement $x$. | | $K.E. = \frac{1}{2} k (x_o^2 - x^2)$ | Kinetic Energy at displacement $x$. | | $T.E. = \frac{1}{2} k x_o^2$ | Total Energy of the system. | | $T.E. = \frac{1}{2} m \omega^2 x_o^2$ | Total Energy (alternative form). | This principle is fundamental to understanding oscillatory systems, from simple pendulums to molecular vibrations. For more on energy in fields, see .