Problem Statement
A body rests on a rough cone with vertical axis and half-angle $\alpha$. The cone rotates with angular velocity $\omega$. Coefficient of friction $\mu$. Find the range of $\omega$ for the body to remain stationary at distance $r$ from axis.
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 body rests on a rough cone with vertical axis and half-angle $\alpha$. The cone rotates with angular velocity $\omega$. Coefficient of friction $\mu$. Find the range of $\omega$ for the body to remain stationary at distance $r$ from axis.
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 body rests on a rough cone with vertical axis and half-angle $\alpha$. The cone rotates with angular velocity $\omega$. Coefficient of friction $\mu$. Find the range of $\omega$ for the body to remain stationary at distance $r$ from axis.
Given Information
- Half-angle: $\alpha$, friction: $\mu$, radius from axis: $r$
Physical Concepts & Formulas
Forces: gravity $mg$ down, normal $N$ perpendicular to cone face, friction $f \leq \mu N$ along the face. Two limiting cases: friction pointing down the slope ($\omega_{\max}$, body tends to slide up) and up the slope ($\omega_{\min}$, body tends to slide down).
Step-by-Step Solution
Step 1 — For $\omega_{\max}$ (friction downward along slope):
Vertical: $N\cos\alpha – \mu N\sin\alpha = mg$
Horizontal: $N\sin\alpha + \mu N\cos\alpha = m\omega_{\max}^2 r$
Dividing: $\omega_{\max}^2 = \dfrac{g(\tan\alpha+\mu)}{r(1-\mu\tan\alpha)}$
Step 2 — For $\omega_{\min}$ (friction upward along slope):
$$\omega_{\min}^2 = \frac{g(\tan\alpha-\mu)}{r(1+\mu\tan\alpha)}$$
Answer
$$\boxed{\sqrt{\frac{g(\tan\alpha-\mu)}{r(1+\mu\tan\alpha)}} \le \omega \le \sqrt{\frac{g(\tan\alpha+\mu)}{r(1-\mu\tan\alpha)}}}$$
Physical Interpretation
A band of safe angular velocities exists between $\omega_{\min}$ and $\omega_{\max}$. Greater friction widens this band. If $\mu \geq \tan\alpha$, the body stays at rest even without rotation ($\omega_{\min}=0$).
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{\sqrt{\frac{g(\tan\alpha-\mu)}{r(1+\mu\tan\alpha)}} \le \omega \le \sqrt{\frac{g(\tan\alpha+\mu)}{r(1-\mu\tan\alpha)}}}}$$
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
$$\boxed{\boxed{\boxed{\sqrt{\frac{g(\tan\alpha-\mu)}{r(1+\mu\tan\alpha)}} \le \omega \le \sqrt{\frac{g(\tan\alpha+\mu)}{r(1-\mu\tan\alpha)}}}}}$$
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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