Problem Statement
A rigid body (mass $m$) pivots at distance $d$ from its CM. Find the period and equivalent simple-pendulum length.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: With $I=m(k^2+d^2)$ (parallel-axis theorem, $k$ = radius of gyration):
Step 2 — Apply the relevant physical law or equation: $$T = 2\pi\sqrt{\frac{I}{mgd}} = 2\pi\sqrt{\frac{k^2+d^2}{gd}}$$
$$\boxed{l_{\rm eq} = \frac{k^2+d^2}{d}, \quad T_{\min} = 2\pi\sqrt{\frac{2k}{g}} \text{ at } d=k}$$
Step 3 — Solve algebraically for the unknown: Pivot and centre of oscillation (distance $l_{\rm eq}$ below pivot) are interchangeable — Kater’s pendulum principle.
Worked Calculation
$$T = 2\pi\sqrt{\frac{I}{mgd}} = 2\pi\sqrt{\frac{k^2+d^2}{gd}}$$
$$\boxed{l_{\rm eq} = \frac{k^2+d^2}{d}, \quad T_{\min} = 2\pi\sqrt{\frac{2k}{g}} \text{ at } d=k}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
With $I=m(k^2+d^2)$ (parallel-axis theorem, $k$ = radius of gyration):
$$T = 2\pi\sqrt{\frac{I}{mgd}} = 2\pi\sqrt{\frac{k^2+d^2}{gd}}$$
$$\boxed{l_{\rm eq} = \frac{k^2+d^2}{d}, \quad T_{\min} = 2\pi\sqrt{\frac{2k}{g}} \text{ at } d=k}$$
Pivot and centre of oscillation (distance $l_{\rm eq}$ below pivot) are interchangeable — Kater’s pendulum principle.
Answer
$$\boxed{l_{\rm eq} = \frac{k^2+d^2}{d}, \quad T_{\min} = 2\pi\sqrt{\frac{2k}{g}} \text{ at } d=k}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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