Simple Pendulum: Forces, Tension & Energy Conservation

Interactive Simple Pendulum Simulation: A Guide to Vector Forces & Dynamics

The Simple Pendulum Use this simulation to analyze vector forces in real-time.

1. Controls

• Parameters: Adjust Initial Angle (θ₀), Length (L), and Mass (m) on the right. Note that angles > 15° deviate from Simple Harmonic Motion.
• Playback: Use Start/Pause to toggle motion. Use Step to freeze frame at specific points like the bottom (equilibrium).

2. Key Physics Concepts

1. Restoring Force: In Gravity Mode, watch the tangential arrow (mg sinθ). It always points to the centre. It vanishes at the bottom (equilibrium) and is largest at the top (max amplitude), indicating where acceleration (α) is highest.
2. Tension vs. Gravity: In All Mode, pause at the bottom. Notice the Red Tension arrow (T) is longer than the Green Gravity arrow (mg). ◦ Formula: T = mg + mv²/L ◦ Tension must support the weight plus provide the centripetal force to curve the path.
3. Mass Independence: Doubling Mass (m) doubles the forces (T and mg) but the Period (T) and angular frequency (ω) stay the same. Inertia cancels out the increased gravitational pull.

When does a pendulum behave like simple harmonic motion?

In short: Gravity splits into a radial part mg cos θ along the string and a tangential part mg sin θ that always points back toward the lowest point. The tangential part is the restoring force. Only for small angles does sin θ ≈ θ, which makes the restoring force proportional to the displacement and gives simple harmonic motion with T = 2π√(L ÷ g). At larger angles the motion deviates from SHM and the real period is longer.

Common doubts

• “Does the mass of the bob change the period?” No. For small angles T = 2π√(L ÷ g) depends on the length, not on the mass.
• “Is the string tension just mg?” No. The tension is FT = mg cos θ + mv² ÷ L, so it is larger than mg when the bob is swinging through the bottom, because it must also supply the centripetal force.
• “Why does the graph show a dashed small-angle curve?” It is the SHM prediction. It matches the real motion for small angles and drifts away from it as the starting angle grows.

Change the starting angle, length and mass in the simulation above, press START, and compare the actual motion with the small-angle prediction.

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