Problem Statement
A simple pendulum of length $l$ is in a lift. Find the period when the lift (a) accelerates upward with $w$, (b) accelerates downward with $w$, (c) falls freely.
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 simple pendulum of length $l$ is in a lift. Find the period when the lift (a) accelerates upward with $w$, (b) accelerates downward with $w$, (c) falls freely.
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 simple pendulum of length $l$ is in a lift. Find the period when the lift (a) accelerates upward with $w$, (b) accelerates downward with $w$, (c) falls freely.
Given Information
- Pendulum length: $l$, lift acceleration: $w$
Physical Concepts & Formulas
In the non-inertial frame of the lift, a pseudo-force acts on the pendulum bob. Effective gravity changes: $g_{\rm eff} = g \pm w$. Period: $T = 2\pi\sqrt{l/g_{\rm eff}}$.
Step-by-Step Solution
(a) Lift accelerates upward: $g_{\rm eff} = g+w$
$$T_a = 2\pi\sqrt{\frac{l}{g+w}}$$
(b) Lift accelerates downward: $g_{\rm eff} = g-w$
$$T_b = 2\pi\sqrt{\frac{l}{g-w}}$$
(c) Free fall: $g_{\rm eff} = 0$, period $T_c = \infty$ (no oscillation).
Answer
$$\boxed{T_a = 2\pi\sqrt{\frac{l}{g+w}},\quad T_b = 2\pi\sqrt{\frac{l}{g-w}},\quad T_c = \infty}$$
Physical Interpretation
An upward-accelerating lift feels heavier — the pendulum swings faster. A downward-accelerating lift feels lighter — it swings slower. In free fall (Einstein elevator), effective gravity vanishes and the pendulum has no restoring force. Astronauts on the ISS cannot use pendulum clocks.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{T_a = 2\pi\sqrt{\frac{l}{g+w}},\quad T_b = 2\pi\sqrt{\frac{l}{g-w}},\quad T_c = \infty}}$$
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
$$\boxed{\boxed{\boxed{T_a = 2\pi\sqrt{\frac{l}{g+w}},\quad T_b = 2\pi\sqrt{\frac{l}{g-w}},\quad T_c = \infty}}}$$
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.
Leave a Reply