Precision Gravity Lab: Kater's Pendulum
Use the **Kater's Pendulum** method—a reversible physical pendulum—to precisely determine the value of **acceleration due to gravity ($g$)**. This technique is highly accurate as it measures the **compound pendulum's** equivalent length.
Key Equations & Concepts
▼The acceleration due to gravity is calculated when the periods from both knife edges ($T_1$ and $T_2$) are equal ($T_1 \approx T_2$): $$ g = \frac{8\pi^2 (l_1 - l_2)}{T_1^2 - T_2^2} \quad \text{or} \quad g = \frac{4\pi^2 (l_1 + l_2)}{T_1^2 + T_2^2} $$ In an ideal, perfectly adjusted system ($T_1=T_2$), it simplifies to: $$ g = \frac{4\pi^2 L_{\text{equiv}}}{T^2} \quad \text{where } L_{\text{equiv}} = \frac{l_1 + l_2}{2} $$
We use the **equivalent length** $L_{\text{equiv}} = \frac{l_1 + l_2}{2}$ in the challenge, assuming the periods are nearly equal.
A Kater's pendulum is adjusted until its periods of oscillation about the two **knife edges** ($K_1$ and $K_2$) are equal. When $T_1 = T_2$, the distance between $K_1$ and $K_2$ is equal to the **length of an equivalent simple pendulum**.
Experiment 1: Period Measurement and Equalization
Simulate adjusting the mass positions to achieve equal periods ($T_1 \approx T_2$).
Adjust M1 to equalize $T_1$ and $T_2$ (Ideal $T_1=T_2$).
Experiment 2: Precise $g$ Calculation Challenge
Calculate $g$ using the simplified ideal formula $g = \frac{4\pi^2 L_{\text{equiv}}}{T^2}$ with the given measured values.
A **physical pendulum** (or compound pendulum, like Kater's) has its mass distributed, unlike a simple pendulum which assumes all mass is concentrated at a point. Kater's method cleverly uses the periods from two pivot points to find the **equivalent length ($L_{\text{equiv}}$)** of a simple pendulum.
Rotational Dynamics and Precision
The time period of a physical pendulum is $T = 2\pi \sqrt{\frac{k^2 + h^2}{gh}}$, where $h$ is the distance from the pivot to the center of mass ($CM$). Kater's pendulum eliminates the need to find the $CM$ or $k$ directly, making it highly accurate.
In real experiments, **air resistance** and **friction** cause the amplitude to decrease (damping). Precision measurements require timing a large number of oscillations and dividing to get the average period.
