Total Energy of SHM

Introduction

Key Concepts

1. Definition of Total Energy in SHM

In Simple Harmonic Motion, the total mechanical energy remains constant if there is no non-conservative force acting on the system, such as friction or air resistance. This total energy is the sum of kinetic energy (KE) and potential energy (PE). Mathematically, it is expressed as: $$E_{total} = KE + PE$$

2. Kinetic Energy in SHM

Kinetic energy in SHM varies with the displacement of the oscillating object. It is given by: $$KE = \frac{1}{2}mv^2$$ where \( m \) is the mass and \( v \) is the velocity of the object. Since velocity varies with time in SHM, kinetic energy also oscillates between maximum and minimum values.

3. Potential Energy in SHM

Potential energy in SHM is associated with the displacement from the equilibrium position. For a mass-spring system, it is defined as: $$PE = \frac{1}{2}kx^2$$ where \( k \) is the spring constant and \( x \) is the displacement. Potential energy is maximum when the displacement is maximum and zero at the equilibrium position.

4. Mathematical Representation of SHM

SHM can be described using sinusoidal functions. The displacement \( x(t) \) as a function of time \( t \) is: $$x(t) = A \cos(\omega t + \phi)$$ where:

• \( A \) = amplitude
• \( \omega \) = angular frequency
• \( \phi \) = phase constant

5. Total Mechanical Energy in SHM

Since energy is conserved in SHM (in the absence of non-conservative forces), the total mechanical energy remains constant. By substituting the expressions for KE and PE, we get: $$E_{total} = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$$ Substituting \( v(t) \) from the velocity equation: $$E_{total} = \frac{1}{2}m(A\omega)^2 \sin^2(\omega t + \phi) + \frac{1}{2}k(A^2 \cos^2(\omega t + \phi))$$ Using the relationship \( k = m\omega^2 \): $$E_{total} = \frac{1}{2}kA^2 (\sin^2(\omega t + \phi) + \cos^2(\omega t + \phi))$$ Since \( \sin^2 \theta + \cos^2 \theta = 1 \): $$E_{total} = \frac{1}{2}kA^2$$ This shows that the total energy is solely dependent on the amplitude and the spring constant.

6. Energy Conservation in SHM

Energy conservation implies that energy continuously transforms between kinetic and potential forms while the total remains constant. At maximum displacement (\( x = A \)):

• Potential Energy (\( PE_{max} \)) = \( \frac{1}{2}kA^2 \)
• Kinetic Energy (\( KE \)) = 0

• Potential Energy (\( PE \)) = 0
• Kinetic Energy (\( KE_{max} \)) = \( \frac{1}{2}kA^2 \)

7. Phase Relationship Between KE and PE

Kinetic and potential energies are out of phase by \( 90^\circ \) in SHM. When KE is maximum, PE is zero, and vice versa. This phase difference ensures the continuous transfer of energy, crucial for sustained oscillations.

8. Energy in Different SHM Systems

While the mass-spring system is a common example, SHM can also describe other systems like pendulums (for small angles), electrical oscillators, and molecular vibrations. In each case, the total energy formulation remains similar, with appropriate variables replacing mass, spring constant, and displacement.

9. Damping and Its Effect on Total Energy

In real-world scenarios, damping forces like friction cause the total mechanical energy to decrease over time. The energy dissipates as heat, leading to a gradual reduction in amplitude. The equation for total energy with damping is: $$E(t) = \frac{1}{2}kA^2 e^{-2bt/m}$$ where \( b \) is the damping coefficient and \( m \) is the mass. This exponential decay signifies energy loss due to non-conservative forces.

10. Calculating Total Energy with Given Parameters

Consider a mass-spring system with mass \( m = 2 \, \text{kg} \), spring constant \( k = 50 \, \text{N/m} \), and amplitude \( A = 0.1 \, \text{m} \). The total energy is: $$E_{total} = \frac{1}{2}kA^2 = \frac{1}{2} \times 50 \times (0.1)^2 = 0.5 \, \text{J}$$ This calculation demonstrates how amplitude and spring constant directly influence the total energy in SHM.

Comparison Table

Summary and Key Takeaways