Problem Statement
A particle of mass $m$ rests at the center of a smooth horizontal tube of length $2l$ rotating at constant angular velocity $\omega$ about its midpoint. The particle is released from rest (relative to tube). Find its speed relative to the tube when it reaches the end.
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 particle of mass $m$ rests at the center of a smooth horizontal tube of length $2l$ rotating at constant angular velocity $\omega$ about its midpoint. The particle is released from rest (relative to tube). Find its speed relative to the tube when it reaches the end.
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 particle of mass $m$ rests at the center of a smooth horizontal tube of length $2l$ rotating at constant angular velocity $\omega$ about its midpoint. The particle is released from rest (relative to tube). Find its speed relative to the tube when it reaches the end.
Given Information
- Tube half-length: $l$, angular velocity: $\omega$ (constant), starts at rest at center
Physical Concepts & Formulas
In the rotating frame, centrifugal force $m\omega^2 r$ drives the particle outward. Equation of motion: $m\ddot{r} = m\omega^2 r$, which gives exponential growth.
Step-by-Step Solution
Step 1 — Multiply by $\dot{r}$:
$$\dot{r}\ddot{r} = \omega^2 r\dot{r} \implies \frac{d}{dt}\left(\frac{\dot{r}^2}{2}\right) = \frac{d}{dt}\left(\frac{\omega^2 r^2}{2}\right)$$
$$\dot{r}^2 = \omega^2 r^2 \quad (\text{since } \dot{r}=r=0 \text{ at } t=0)$$
$$\dot{r} = \omega r$$
Step 2 — Speed at $r = l$:
$$v_{\rm rel} = \omega l$$
Answer
$$\boxed{v_{\rm rel} = \omega l}$$
Physical Interpretation
The radial speed at exit equals the tangential speed of the tube end — a beautiful result. The particle also has tangential speed $\omega l$ from the tube’s rotation, so its total lab-frame speed at exit is $\omega l\sqrt{2}$, directed at $45°$ outward from the radial direction.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{v_{\rm rel} = \omega l}}$$
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
$$\boxed{\boxed{\boxed{v_{\rm rel} = \omega l}}}$$
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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