Displacement, Velocity, and Acceleration in Simple Harmonic Motion

Introduction

Key Concepts

1. Displacement in Simple Harmonic Motion

Displacement is the measure of the distance and direction of an object from its mean or equilibrium position. In SHM, displacement varies sinusoidally with time, characterized by its amplitude, frequency, and phase.

The displacement ($x$) of an object in SHM can be described by the equation:

$$x(t) = A \cos(\omega t + \phi)$$

Where:

• $A$ is the amplitude, the maximum displacement from the equilibrium position.
• $\omega$ is the angular frequency, related to the period ($T$) by $\omega = \frac{2\pi}{T}$.
• $\phi$ is the phase constant, determining the object's position at $t = 0$.

**Example:** Consider a mass-spring system oscillating with an amplitude of 5 cm, a period of 2 seconds, and starting from the maximum displacement. The displacement as a function of time is:

$$x(t) = 5 \cos\left(\frac{2\pi}{2} t + 0\right) = 5 \cos(\pi t) \text{ cm}$$

2. Velocity in Simple Harmonic Motion

Velocity in SHM is the rate of change of displacement with respect to time. It indicates how fast the object is moving and its direction at any given moment.

The velocity ($v$) is the first derivative of displacement with respect to time:

$$v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi)$$

Alternatively, velocity can be expressed as:

$$v(t) = A \omega \cos\left(\omega t + \phi + \frac{\pi}{2}\right)$$

This shows that velocity is also a sinusoidal function, but it is phase-shifted by $\frac{\pi}{2}$ radians compared to displacement.

**Example:** Using the previous displacement equation, the velocity is:

$$v(t) = -5 \times \pi \sin(\pi t) \text{ cm/s} = -5\pi \sin(\pi t) \text{ cm/s}$$

3. Acceleration in Simple Harmonic Motion

Acceleration in SHM is the rate of change of velocity with respect to time. It is directly proportional to displacement but acts in the opposite direction, indicating a restoring force.

The acceleration ($a$) is the first derivative of velocity or the second derivative of displacement:

$$a(t) = \frac{dv}{dt} = -A \omega^2 \cos(\omega t + \phi)$$ $$a(t) = -\omega^2 x(t)$$

**Example:** Continuing with the mass-spring system, the acceleration is:

$$a(t) = -5 \times \pi^2 \cos(\pi t) \text{ cm/s}^2 = -5\pi^2 \cos(\pi t) \text{ cm/s}^2$$

4. Phase Relationships

In SHM, displacement, velocity, and acceleration are out of phase with each other:

• **Displacement and Velocity:** Velocity leads displacement by $\frac{\pi}{2}$ radians.
• **Velocity and Acceleration:** Acceleration leads velocity by $\frac{\pi}{2}$ radians.
• **Displacement and Acceleration:** Acceleration leads displacement by $\pi$ radians (i.e., they are in opposite directions).

5. Energy in Simple Harmonic Motion

Energy in SHM oscillates between kinetic and potential forms. The total mechanical energy remains constant in the absence of non-conservative forces.

The kinetic energy ($KE$) is given by:

$$KE = \frac{1}{2}mv^2 = \frac{1}{2}m(A\omega)^2 \sin^2(\omega t + \phi)$$

The potential energy ($PE$) stored in the restoring force is:

$$PE = \frac{1}{2}kx^2 = \frac{1}{2}kA^2 \cos^2(\omega t + \phi)$$

Where $k$ is the spring constant related to angular frequency by $k = m\omega^2$.

6. Period and Frequency

The period ($T$) is the time taken for one complete cycle of oscillation, and frequency ($f$) is the number of cycles per second.

The relationship between period and angular frequency is:

$$\omega = \frac{2\pi}{T}$$ $$f = \frac{1}{T}$$

**Example:** If a pendulum has a period of 2 seconds, its angular frequency is:

$$\omega = \frac{2\pi}{2} = \pi \text{ rad/s}$$ $$f = \frac{1}{2} \text{ Hz}$$

7. Damping and Resonance

While ideal SHM assumes no energy loss, real systems experience damping due to friction or resistance, causing amplitude to decrease over time. Resonance occurs when a system is driven at its natural frequency, leading to large amplitude oscillations.

The equation of motion with damping is:

$$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$$

Where $b$ is the damping coefficient. The solution depends on the damping ratio, determining whether the system is underdamped, overdamped, or critically damped.

8. Applications of SHM

SHM principles are applied in various fields, including engineering, astronomy, and everyday devices:

• Mass-Spring Systems: Used to model vibrations in buildings and vehicles.
• Pendulums: Utilized in clocks to maintain accurate timekeeping.
• Electrical Circuits: LC circuits exhibit oscillatory behavior analogous to SHM.
• Wave Motion: Understanding SHM is essential for analyzing sound waves and light waves.

Advanced Concepts

1. Mathematical Derivation of SHM Equations

To derive the fundamental equations of SHM, consider Newton's second law applied to a restoring force proportional to displacement:

$$F = -kx$$ $$ma = -kx$$ $$m\frac{d^2x}{dt^2} + kx = 0$$

This differential equation has solutions of the form:

$$x(t) = A \cos(\omega t + \phi)$$

Where $\omega = \sqrt{\frac{k}{m}}$ is the angular frequency. Taking derivatives, we obtain velocity and acceleration:

$$v(t) = -A \omega \sin(\omega t + \phi)$$ $$a(t) = -A \omega^2 \cos(\omega t + \phi)$$

2. Energy Conservation in SHM

In the absence of damping, mechanical energy in SHM is conserved. The total energy ($E$) is the sum of kinetic and potential energies:

$$E = KE + PE = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$$

Substituting the expressions for velocity and displacement:

$$E = \frac{1}{2}m(A\omega)^2 \sin^2(\omega t + \phi) + \frac{1}{2}kA^2 \cos^2(\omega t + \phi)$$ $$E = \frac{1}{2}mA^2\omega^2 (\sin^2(\omega t + \phi) + \cos^2(\omega t + \phi))$$ $$E = \frac{1}{2}mA^2\omega^2$$

Since $\omega^2 = \frac{k}{m}$:

$$E = \frac{1}{2}kA^2$$

This shows that the total energy depends only on the amplitude and the spring constant, remaining constant over time.

3. Complex Problem-Solving in SHM

**Problem:** A mass-spring system with mass $m = 2 \text{ kg}$ and spring constant $k = 50 \text{ N/m}$ is released from rest at a displacement of $A = 0.1 \text{ m}$. Determine the velocity and acceleration at $t = 0.1 \text{ s}$.

**Solution:**

First, find the angular frequency:

$$\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5 \text{ rad/s}$$

Displacement as a function of time:

$$x(t) = A \cos(\omega t) = 0.1 \cos(5 \times 0.1) = 0.1 \cos(0.5) \text{ m}$$

Calculate $\cos(0.5)$ (radians):

$$\cos(0.5) \approx 0.8776$$ $$x(0.1) = 0.1 \times 0.8776 = 0.08776 \text{ m}$$

Velocity:

$$v(t) = -A \omega \sin(\omega t) = -0.1 \times 5 \sin(0.5)$$ $$v(0.1) = -0.5 \times 0.4794 \approx -0.2397 \text{ m/s}$$

Acceleration:

$$a(t) = -\omega^2 x(t) = -25 \times 0.08776 \approx -2.194 \text{ m/s}^2$$

**Answer:** At $t = 0.1 \text{ s}$, the velocity is approximately $-0.2397 \text{ m/s}$ and the acceleration is approximately $-2.194 \text{ m/s}^2$.

4. Phase Space Representation

Phase space is a graphical representation of the state of a system, plotting displacement against momentum or velocity. For SHM, the trajectory in phase space is an ellipse, reflecting the sinusoidal nature of displacement and velocity.

Using displacement ($x$) and velocity ($v$):

$$\left(\frac{x}{A}\right)^2 + \left(\frac{v}{A\omega}\right)^2 = \cos^2(\omega t + \phi) + \sin^2(\omega t + \phi) = 1$$

This equation represents an ellipse with semi-major axis $A$ and semi-minor axis $A\omega$.

5. Damped Simple Harmonic Motion

In real-world scenarios, oscillatory systems experience damping due to energy loss mechanisms like friction or air resistance. The equation of motion for damped SHM is:

$$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$$

Where $b$ is the damping coefficient. The solution depends on the damping ratio ($\zeta$):

$$\zeta = \frac{b}{2\sqrt{mk}}$$

Depending on $\zeta$, the system can be:

• Underdamped ($\zeta < 1$): Oscillatory motion with exponentially decreasing amplitude.
• Overdamped ($\zeta > 1$): Non-oscillatory motion returning to equilibrium.
• Critically damped ($\zeta = 1$): Returns to equilibrium in the shortest time without oscillating.

**Example:** For a mass $m = 1 \text{ kg}$ and spring constant $k = 100 \text{ N/m}$, calculate the damping coefficient for critical damping.

$$\zeta = 1 = \frac{b}{2\sqrt{1 \times 100}}$$ $$b = 2 \times \sqrt{100} = 20 \text{ Ns/m}$$

6. Driven Simple Harmonic Motion and Resonance

Driven SHM occurs when an external periodic force acts on the system. The equation of motion becomes:

$$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0 \cos(\omega_{\text{drive}} t)$$

Resonance occurs when the driving frequency ($\omega_{\text{drive}}$) matches the system's natural frequency ($\omega_0$), leading to maximum amplitude oscillations:

$$\omega_{\text{drive}} = \omega_0 = \sqrt{\frac{k}{m}}$$

**Applications:** Resonance phenomena are utilized in musical instruments, radio tuners, and bridge engineering to enhance performance or ensure stability.

7. Interdisciplinary Connections

SHM principles extend beyond physics into various disciplines:

• Engineering: SHM is critical in designing structures that can withstand oscillatory forces, such as bridges and skyscrapers.
• Biology: Models of SHM describe heartbeats and lung functions.
• Medicine: Understanding SHM aids in developing medical devices like pacemakers.
• Economics: Oscillatory models help analyze business cycles and market fluctuations.

8. Nonlinear Simple Harmonic Motion

In real systems, restoring forces may not always be proportional to displacement, leading to nonlinear SHM. This results in amplitude-dependent frequencies and more complex motion dynamics, often requiring numerical methods for analysis.

**Example:** A pendulum with large angular displacement exhibits nonlinear SHM, where the simple harmonic approximation breaks down, and higher-order terms in the expansion are necessary for accurate modeling.

Comparison Table

Aspect	Displacement	Velocity	Acceleration
Definition	Position relative to equilibrium.	Rate of change of displacement.	Rate of change of velocity.
Equation	$x(t) = A \cos(\omega t + \phi)$	$v(t) = -A \omega \sin(\omega t + \phi)$	$a(t) = -A \omega^2 \cos(\omega t + \phi)$
Phase Relationship	Reference phase.	Leads displacement by $\frac{\pi}{2}$ radians.	Leads displacement by $\pi$ radians.
Energy Contribution	Potential Energy ($PE$).	Kinetic Energy ($KE$).	Linked to the restoring force; $PE$ and $KE$ depend on displacement and velocity.
Physical Interpretation	Extent of oscillation.	Speed of oscillating object.	Force acting to restore equilibrium.

Summary and Key Takeaways