Problem Statement
A particle moves in a circle of radius $R$ with initial speed $v_0$. Tangential deceleration is constant: $w_\tau = a$. Find the total arc length traveled before the particle stops and the number of revolutions.
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 moves in a circle of radius $R$ with initial speed $v_0$. Tangential deceleration is constant: $w_\tau = a$. Find the total arc length traveled before the particle stops and the number of revolutions.
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 moves in a circle of radius $R$ with initial speed $v_0$. Tangential deceleration is constant: $w_\tau = a$. Find the total arc length traveled before the particle stops and the number of revolutions.
Given Information
- Initial speed: $v_0$, tangential deceleration: $a$, radius: $R$
Physical Concepts & Formulas
For tangential deceleration, the arc length calculation is identical to linear motion deceleration. The radius only determines how many full circles are completed.
Step-by-Step Solution
Step 1 — Stopping distance (arc length): $v^2 = v_0^2 – 2as$
At $v=0$: $s = v_0^2/(2a)$
Step 2 — Number of revolutions:
$$N = \frac{s}{2\pi R} = \frac{v_0^2}{4\pi a R}$$
Answer
$$\boxed{s = \frac{v_0^2}{2a},\quad N = \frac{v_0^2}{4\pi aR}}$$
Physical Interpretation
The total arc length to stop is identical to straight-line braking — the curve shape doesn’t matter. This means a skater braking on a circular rink covers the same arc length as one braking in a straight line with the same deceleration. The number of revolutions decreases for larger radius (fewer full loops before stopping).
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{s = \frac{v_0^2}{2a},\quad N = \frac{v_0^2}{4\pi aR}}}$$
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
$$\boxed{\boxed{\boxed{s = \frac{v_0^2}{2a},\quad N = \frac{v_0^2}{4\pi aR}}}}$$
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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