Circular Motion & Centripetal Force

Circular motion occurs when a body moves along a circular path. It requires a continuous inward (centripetal) force to maintain the curved trajectory. Without this force, the body would fly off tangentially due to inertia — there is no real outward "centrifugal force" in an inertial frame.

Uniform Circular Motion (UCM): The speed is constant, but velocity continuously changes direction.
The centripetal acceleration is $a_{c}$ = $v^{2}$/r = r*$\omega$², always directed toward the centre.
The centripetal force is $F_{c}$ = $mv^{2}$/r = mr*$\omega$².
Angular velocity $\omega$ = 2*$\pi$f = 2$\pi$/T, where f is frequency and T is the time period.

Key relationships: v = r*$\omega$, $a_{c}$ = v*$\omega$ = $v^{2}$/r = r*$\omega$². The velocity is tangential, and the acceleration is radial (toward centre). Since the force is perpendicular to velocity, the centripetal force does zero work.

Non-Uniform Circular Motion: When speed changes along the circular path, there are two components of acceleration:

• Centripetal (radial): $a_{c}$ = $v^{2}$/r (toward centre)
• Tangential: $a_{t}$ = dv/dt = r*$\alpha$ (along the tangent)
• Net acceleration: a = $\sqrt{{a}_{c}^2 + {a}_{t}^2}$, at angle $\theta$ = arctan($\frac{a_{t}}{a_{c}}$) from the radius.

Vertical Circle: A classic non-uniform circular motion problem. For a mass on a string of length L:

• At the top: $T_{top}$ + mg = $mv_{top}^{2}$/L, so $T_{top}$ = $mv_{top}^{2}$/L - mg.
  The minimum speed at the top ($T_{top}$ = 0) gives $v_{top_min}$ = $\sqrt{gL}$.
• At the bottom: $T_{bottom}$ - mg = $mv_{bottom}^{2}$/L, so $T_{bottom}$ = $mv_{bottom}^{2}$/L + mg.
• Using energy conservation: $v_{bottom}^{2}$ = $v_{top}^{2}$ + 4gL.
  Minimum speed at bottom for complete loop: $v_{bottom_min}$ = $\sqrt{5gL}$.
• Tension difference: $T_{bottom}$ - $T_{top}$ = 6mg (always, regardless of speed).

For a mass on a rigid rod, the minimum speed at the top is zero (the rod can push), so $v_{bottom_min}$ = $\sqrt{4gL}$ = 2*$\sqrt{gL}$.

Banked Road: A vehicle on a banked road of angle $\theta$ and radius r:

• Without friction: tan($\theta$) = $v^{2}$/(rg). The speed is fixed for a given bank angle.
• With friction: Maximum speed: $v_{max}$ = $\sqrt{rg(tan(\theta}$ + $\mu$)/(1 - $\mu$*tan($\theta$))); Minimum speed: $v_{min}$ = $\sqrt{rg(tan(\theta}$ - $\mu$)/(1 + $\mu$*tan($\theta$))).

Conical Pendulum: A mass m on a string of length L making angle $\theta$ with the vertical.
Tsin($\theta$) = m$\omega$²r = m$\omega$²Lsin($\theta$), giving Tcos($\theta$) = mg.
Time period: $T_{p}$ = 2$\pi$*$\sqrt{L*cos(\theta}$/g).

The key problem-solving concept is: identify all real forces (gravity, tension, normal, friction), resolve them into radial and tangential components, and set the net radial force equal to $mv^{2}$/r — never add a fictitious centrifugal force in an inertial frame.

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