Problem Statement
A Foucault pendulum oscillates at latitude $\lambda$. Find the period of precession of the pendulum’s oscillation plane due to Earth’s rotation.
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
A Foucault pendulum oscillates at latitude $\lambda$. Find the period of precession of the pendulum’s oscillation plane due to Earth’s rotation.
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
A Foucault pendulum oscillates at latitude $\lambda$. Find the period of precession of the pendulum’s oscillation plane due to Earth’s rotation.
Given Information
- Latitude: $\lambda$, Earth’s rotation period: $T_E = 24$ h, $\Omega = 2\pi/T_E$
Physical Concepts & Formulas
The vertical component of Earth’s angular velocity ($\Omega\sin\lambda$) causes the pendulum plane to precess. The precession rate equals $\Omega_{\rm prec} = \Omega\sin\lambda$, so the period of precession is $T_{\rm prec} = 2\pi/(\Omega\sin\lambda) = T_E/\sin\lambda$.
Step-by-Step Solution
Step 1 — Coriolis force on pendulum bob: For horizontal velocity $v$ of bob, horizontal Coriolis component: $2m\Omega\sin\lambda\cdot v$ (perpendicular to $v$).
Step 2 — This acts like a pendulum on a rotating platform with angular velocity $\Omega_z = \Omega\sin\lambda$.
Step 3 — Precession period:
$$T_{\rm prec} = \frac{2\pi}{\Omega\sin\lambda} = \frac{24\text{ h}}{\sin\lambda}$$
Worked Calculation
At Paris ($\lambda \approx 49°$): $T_{\rm prec} = 24/\sin49° = 24/0.755 \approx 31.8$ h
At pole ($\lambda = 90°$): $T_{\rm prec} = 24$ h (one full rotation per day)
At equator: $T_{\rm prec} = \infty$ (no precession)
Answer
$$\boxed{T_{\rm prec} = \frac{24\text{ h}}{\sin\lambda}}$$
Physical Interpretation
Foucault demonstrated Earth’s rotation in 1851 at the Panthéon in Paris with a 67-meter pendulum. The plane of oscillation rotates slowly, completing a full $360°$ rotation in $24/\sin\lambda$ hours. At the poles, the period is exactly 24 hours. At the equator, the Coriolis effect on a pendulum vanishes — no precession occurs.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{T_{\rm prec} = \frac{24\text{ h}}{\sin\lambda}}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{\boxed{T_{\rm prec} = \frac{24\text{ h}}{\sin\lambda}}}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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