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 (a_c)

\[
a_c = \frac{v^2}{r} = \omega^2 r
\]
Where:

• \(v\): Tangential velocity (m/s)
• \(\omega\): Angular velocity (rad/s)
• \(r\): Radius (m)

Centripetal Force (F_c)

\[
F_c = m\frac{v^2}{r} = m\omega^2 r
\]
Force Sources:

• Tension (pendulums)
• Friction (vehicle turns)
• Gravity (orbits)

Advanced Concepts

Banked Curves

Ideal banking angle (\(\theta\)) without friction:
\[
\tan\theta = \frac{v^2}{rg}
\]

Vertical Circular Motion

Tension at top/bottom of loop:
\[
T_{top} = m\left(\frac{v^2}{r} – g\right)
\]
\[
T_{bottom} = m\left(\frac{v^2}{r} + g\right)
\]

Practical Applications

Transportation Engineering

• Highway curves: 8-12° typical banking angles
• High-speed rail: Up to 15° banking

Space Systems

• Geostationary orbit: \(r \approx 42,164\) km
• Centripetal acceleration: \(0.224\) m/s² at ISS altitude

Industrial 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 \(m = 1,500\) kg
• Velocity \(v = 20\) m/s (72 km/h)
• Radius \(r = 50\) m

\[
F_c = \frac{1500 \times 20^2}{50} = 12,000 \, \text{N}
\]
Equivalent to: 1.2 metric tons of force

Common Errors

1. Using diameter instead of radius in calculations
2. Confusing centripetal (center-seeking) with centrifugal (apparent outward) force
3. Neglecting vertical force components in banked turns

Practice Problems

1. A 2,000 kg truck negotiates a 30m radius curve at 15 m/s:
   • Calculate \(a_c\) and \(F_c\)
   • Determine the banking angle for frictionless turning
2. Derive the orbital velocity equation \(v = \sqrt{GM/r}\) from centripetal force
3. Explain how centrifuge RPM relates to separation efficiency.