Simple Pendulum 🕰️
Galileo famously timed the regular swings of a church chandelier with his own pulse and discovered a beautiful rhythm hiding in plain sight. You can feel the same wonder by tying a small stone to a ~100 cm thread and letting it swing! Each gentle to-and-fro takes about 2 s. 🎶 :contentReference[oaicite:0]{index=0}
Set-up 🛠️
- Bob: small mass m
- String: light, inextensible, length L
- Support: rigid point from which the bob hangs
- Angle: θ with the vertical ( θ = 0 at the center)
When you pull the bob sideways a little and release it, it swings along a circular arc of radius L. :contentReference[oaicite:1]{index=1}
Forces in a Swing 💪
Two forces act on the bob:
- Tension T along the string
- Weight mg straight down
Resolve mg into:
- mg cos θ along the string (gives no twist)
- mg sin θ tangential to the arc (pulls the bob back)
The tangential part provides a restoring torque:
$$\tau = -m\,g\,L\,\sin\theta$$
The minus sign just says, “I pull you toward home!” 🏡 :contentReference[oaicite:2]{index=2}
Equation of Motion 📐
Newton’s rotation rule says \( \tau = I\,\alpha \), where I is the moment of inertia and α the angular acceleration. Therefore,
$$I\,\alpha = -m\,g\,L\,\sin\theta$$
Small-Angle Magic ✨
For small swings (up to about 20°) we can safely swap sin θ for θ (in radians):
$$\sin\theta \approx \theta \quad\Rightarrow\quad \alpha = -\frac{m\,g\,L}{I}\,\theta$$
This looks exactly like the simple-harmonic equation, so the pendulum performs simple harmonic motion (SHM) for small angles. :contentReference[oaicite:3]{index=3}
Time Period ⏲️
The SHM equation gives the angular frequency
$$\omega = \sqrt{\frac{m\,g\,L}{I}}\,.$$
For a light string \(I = mL^{2}\), so
$$\omega = \sqrt{\frac{g}{L}},\qquad T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{L}{g}}.$$
Your swing time depends only on the string length and gravity—mass and (small) amplitude do not matter. 🎉 :contentReference[oaicite:4]{index=4}
Quick Example 🌟
Want a “seconds pendulum” that takes exactly 2 s for one full swing?
$$L = \frac{g\,T^{2}}{4\pi^{2}} = \frac{9.8\,\text{m s}^{-2}\,(2\,\text{s})^{2}}{4\pi^{2}} \approx 1\,\text{m}.$$
Hang a 1-m string and you’re ready to tick! ✔️ :contentReference[oaicite:5]{index=5}
Small-Angle Check ✅
A handy table in the text shows that even at 20° the numbers for θ (in radians) and sin θ match closely, confirming why the small-angle shortcut works so well. 📊 :contentReference[oaicite:6]{index=6}
Important NEET Picks 🎯
- The period of a simple pendulum: $$T = 2\pi\sqrt{\dfrac{L}{g}}$$—memorize this gem!
- Small-angle approximation: \( \sin\theta \approx \theta \) (radians) for \( \theta \lesssim 20^\circ \).
- Restoring torque: \( \tau = -m\,g\,L\,\sin\theta \) leads straight to SHM.
- Period depends on length and gravity only—mass and small amplitude play no role.
- Second’s pendulum length at \( g = 9.8\;\text{m s}^{-2} \): \( L \approx 1\;\text{m}. \)
Happy swinging, and may your preparation stay perfectly periodic! 🥳
