Simple Harmonic Motion: Finding ω and Period T

This lesson derives the time period of a simple pendulum by modeling it as Simple Harmonic Motion (SHM). We explore the forces acting on the mass, apply Newton's Second Law, and utilize the small angle approximation to prove that the period depends on length and gravity, but surprisingly, is independent of mass.

What You’ll Learn

1. Identify the forces acting on a pendulum: Tension (T) and Weight (mg).
2. Resolve tension into horizontal and vertical components (T sin θ and T cos θ).
3. Apply the small angle approximation (sin θ ≈ θ and cos θ ≈ 1) to simplify the equations of motion.
4. Derive the formula for angular frequency (ω) and Period (T) of a simple pendulum.
5. Understand why the period is independent of mass and amplitude for small swings.
6. Analyze how changes in gravity (g), such as being on the Moon or in deep space, affect the pendulum's motion.

Key Concepts Covered

Simple Pendulum SHM, Period of Simple Pendulum, Restoring Force, Small Angle Approximation, Angular Frequency (ω), Tension (T), Component Forces, Newton's Second Law.

Why This Lesson Matters

Understanding the simple pendulum is a cornerstone of physics mechanics. It is a classic example of how complex motion can be simplified using linear approximations—a skill essential for AP Physics C, IB Physics, and competitive exams like IIT JEE and NEET. Furthermore, it explains fundamental real-world concepts, such as why pendulum clocks work and how we can measure local gravity.

Prerequisite or Follow-Up Lessons

• Prerequisite: Newton’s Laws of Motion, Vector Resolution
• Follow-Up: Energy in Simple Harmonic Motion, The Physical Pendulum

Full Lesson: Simple Pendulum and Simple Harmonic Motion

1. The Setup and Forces

We start with a mass m (the bob) hanging from a fixed ceiling by a string of length L. When the mass is pulled to the right by a small angle θ, it begins to oscillate. To understand this motion, we must first analyze the forces.

There are two primary forces acting on the mass:

1. Gravity (mg): Pulling straight down.
2. Tension (T): Pulling up along the string towards the pivot.

Instead of tilting our coordinate system, we will use a standard Cartesian system (X and Y axes). We break the Tension (T) into components based on the angle θ:

• Vertical Component: T cos θ (points up, balancing gravity).
• Horizontal Component: T sin θ (points left, acting as the restoring force).

2. Analyzing the X-Direction (The Restoring Force)

Applying Newton’s Second Law to the horizontal (X) direction, we see that the force pulls the mass back toward the center (equilibrium).

T sin θ = m aₓ

Why the minus sign? The minus sign appears because the force is a restoring force. If the displacement x is positive (to the right), the force pulls to the left (negative). This is the mathematical proof that the system seeks equilibrium.

From the geometry of the pendulum, for the triangle formed by the string and the vertical, sin θ is the opposite side over the hypotenuse: sin θ = x / L

Substituting this into our force equation (Equation 1): - T (x / L) = m aₓ

3. Analyzing the Y-Direction and Approximations

Now, let's look at the vertical forces. T cos θ - mg = m aᵧ

Here, we use two crucial approximations to simplify the physics, valid for small angles (typically θ < 10°).

Approximation A: The Small Angle Approximation For small angles, cos θ ≈ 1.

cos 5° ≈ 0.996 (Error ≈ 0.4%)

cos 10° ≈ 0.985 (Error ≈ 1.5%)

Approximation B: Negligible Vertical Acceleration For small oscillations, the mass hardly moves up and down. The vertical acceleration aᵧ is negligible compared to gravity (aᵧ ≈ 0).

Using these approximations in our Y-direction equation: T(1) - mg = 0 which simplifies to: T ≈ mg

Note: Strictly speaking, T varies slightly during the swing (it is higher at the bottom due to centripetal force), but for SHM derivation, treating T ≈ mg is an excellent approximation.

4. Deriving the Equation of Motion

Now, we substitute T = mg back into Equation 1:

(mg) (x / L) = m aₓ

Notice that the mass m appears on both sides. We can cancel it out:

aₓ = - (g / L) x

Does this look familiar? This is the hallmark differential equation for Simple Harmonic Motion: a = - ω² x

By comparing the two equations, we can identify the angular frequency (ω): ω² = g / L Therefore: ω = √(g / L)

5. Calculating the Period (T)

Since the period of motion is defined as T = 2π / ω, we arrive at the famous Period of Simple Pendulum formula:

T = 2π √(L / g)

6. Key Takeaways

1. Independence of Mass: The mass m canceled out during the derivation. A heavy rock and a light feather (ignoring air resistance) will swing with the same period. This happens because gravity pulls harder on heavier objects (mg), but heavier objects are harder to accelerate (ma); these effects cancel perfectly.
2. Independence of Amplitude: For small swings (θ < 10°), the period does not depend on how far you pull the mass back.
3. Dependence on Gravity (g): The period is inversely proportional to √g.
4. On the Moon: Gravity is weaker, so the period is longer (the pendulum swings in "slow motion").
5. In Deep Space: If g = 0, the period is undefined. The pendulum would simply float and would not oscillate because there is no restoring force.