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Circular Motion: Centripetal Dynamics and Engineering Applications
Fundamentals of Circular Motion
An object maintains circular motion when acted upon by a net centripetal force directed toward the rotation center, causing constant change in velocity direction.
Key Equations
Centripetal Acceleration (ac)
![\[ a_c = \frac{v^2}{r} = \omega^2 r \]](https://on22.com/wp-content/ql-cache/quicklatex.com-720a3f1e6b5ed5a19fc0ee9df09733e9_l3.png "Rendered by QuickLaTeX.com")
Where:
![v]( "Rendered by QuickLaTeX.com")
: Tangential velocity (m/s)![\omega]( "Rendered by QuickLaTeX.com")
: Angular velocity (rad/s)![r]( "Rendered by QuickLaTeX.com")
: Radius (m)
: Tangential velocity (m/s)
: Angular velocity (rad/s)
: Radius (m)Centripetal Force (Fc)
![\[ F_c = m\frac{v^2}{r} = m\omega^2 r \]](https://on22.com/wp-content/ql-cache/quicklatex.com-4f469e7977bed213de218eec9515b2f3_l3.png "Rendered by QuickLaTeX.com")
Force Sources:
- Tension (pendulums)
- Friction (vehicle turns)
- Gravity (orbits)
Advanced Concepts
Banked Curves
Ideal banking angle (
) without friction:
![\[ \tan\theta = \frac{v^2}{rg} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-655ad528fc5b766c788016e8e591785e_l3.png "Rendered by QuickLaTeX.com")
Vertical Circular Motion
Tension at top/bottom of loop:
![\[ T_{top} = m\left(\frac{v^2}{r} - g\right) \]](https://on22.com/wp-content/ql-cache/quicklatex.com-71e3b41551a3b1a32b03bc62eb85ea86_l3.png "Rendered by QuickLaTeX.com")
![\[ T_{bottom} = m\left(\frac{v^2}{r} + g\right) \]](https://on22.com/wp-content/ql-cache/quicklatex.com-0cceddef398561a3c3e7c15133c78a6f_l3.png "Rendered by QuickLaTeX.com")
Practical Applications
Transportation Engineering
- Highway curves: 8-12° typical banking angles
- High-speed rail: Up to 15° banking
Space Systems
- Geostationary orbit:
![r \approx 42,164]( "Rendered by QuickLaTeX.com")
km - Centripetal acceleration:
![0.224]( "Rendered by QuickLaTeX.com")
m/s² at ISS altitude
km
m/s² at ISS altitudeIndustrial Technology
- Centrifuges: 10,000-50,000 rpm medical models
- AMOLED screens: Spin coating at 1,500-3,000 rpm
Worked Example
Car on Curved Road:
- Mass
![m = 1,500]( "Rendered by QuickLaTeX.com")
kg - Velocity
![v = 20]( "Rendered by QuickLaTeX.com")
m/s (72 km/h) - Radius
![r = 50]( "Rendered by QuickLaTeX.com")
m
kg
m/s (72 km/h)
m ![\[ F_c = \frac{1500 \times 20^2}{50} = 12,000 \, \text{N} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-b981cf4e16eae5f23a62a99e412dcc22_l3.png "Rendered by QuickLaTeX.com")
Equivalent to: 1.2 metric tons of force
Common Errors
- Using diameter instead of radius in calculations
- Confusing centripetal (center-seeking) with centrifugal (apparent outward) force
- Neglecting vertical force components in banked turns
Practice Problems
- A 2,000 kg truck negotiates a 30m radius curve at 15 m/s:
- Calculate
![a_c]( "Rendered by QuickLaTeX.com")
and ![F_c]( "Rendered by QuickLaTeX.com")
- Determine the banking angle for frictionless turning
- Calculate
- Derive the orbital velocity equation
![v = \sqrt{GM/r}]( "Rendered by QuickLaTeX.com")
from centripetal force - Explain how centrifuge RPM relates to separation efficiency.
and 
from centripetal force
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