Circular Motion

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Circular Motion — Key Facts

Circular motion occurs when a particle moves along a circular path. The motion can be uniform (constant speed) or non-uniform (variable speed). In uniform circular motion, the speed remains constant but the direction of velocity continuously changes — this means there is always an acceleration directed towards the centre, called centripetal acceleration.

Key Formulas:

• Centripetal force: $F_c = \frac{mv^2}{r} = mr\omega^2$
• Centripetal acceleration: $a_c = \frac{v^2}{r} = r\omega^2$
• Angular velocity: $\omega = \frac{2\pi}{T} = 2\pi f$
• Time period: $T = \frac{2\pi r}{v}$
• Frequency: $f = \frac{1}{T}$

Where $m$ = mass, $v$ = linear speed, $r$ = radius, $\omega$ = angular velocity, $T$ = period, $f$ = frequency.

Key facts:

• Centripetal force is NOT a new type of force — it can be tension, friction, gravitational force, or normal reaction acting towards the centre
• The centrifugal force ($mv^2/r$) is a fictitious or pseudo force that appears in a rotating (non-inertial) frame of reference
• For a body moving in a vertical circle (e.g., a roller coaster): minimum speed at the top of the circle to complete the loop: $v_{min} = \sqrt{gr}$

⚡ Exam tip: MDCAT 2023 had a question on banking of roads — the banking angle $\tan\theta = \frac{v^2}{rg}$ is a high-yield formula. Students frequently confuse centripetal with centrifugal force; remember centripetal is real (towards centre), centrifugal is pseudo (appears only in rotating frames). In pulley problems, tension provides the centripetal force.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Circular Motion — Complete Study Guide

Angular Displacement and Velocity: Angular displacement $\theta$ is measured in radians. $\theta = \frac{s}{r}$ where $s$ is arc length. For one complete revolution: $\theta = 2\pi$ rad, $s = 2\pi r$.

Relation between Linear and Angular Quantities:

• $v = r\omega$
• $a_t = r\alpha$ (tangential acceleration)
• $a_c = r\omega^2$

Types of Circular Motion:

Type	Speed	Acceleration
Uniform circular motion	Constant	$a = v^2/r$ towards centre
Non-uniform circular motion	Variable	$a_{total} = \sqrt{a_c^2 + a_t^2}$

Conical Pendulum: A bob moving in a horizontal circle at the end of a string that makes a constant angle $\theta$ with the vertical. Tension $T = \frac{mg}{\cos\theta}$ and horizontal component provides centripetal force: $T\sin\theta = \frac{mv^2}{r}$. Since $r = l\sin\theta$, we get $v = \sqrt{gl\tan\theta}$.

Banking of Roads: Roads are banked to provide the necessary centripetal force without relying on friction alone. The banking angle is given by: $$\tan\theta = \frac{v^2}{rg}$$

Where $v$ is the design speed, $r$ is the radius of curvature.

Vertical Circular Motion: For a body of mass $m$ at the end of a string moving in a vertical circle:

• At the top: minimum tension $T_{min} = 0$ when $v_{top} = \sqrt{gr}$; $T_{top} = \frac{mv^2}{r} - mg$
• At the bottom: $T_{bottom} = \frac{mv^202}{r} + mg$ (maximum tension)
• Minimum speed at the lowest point to complete the circle: $v_{min} = \sqrt{5gr}$

⚡ Common student mistakes: Confusing centripetal with centrifugal force. Using the wrong radius in $v = r\omega$. Forgetting that non-uniform circular motion has both centripetal AND tangential acceleration. In banking questions, students often use the coefficient of friction formula instead of the banking formula.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for thorough preparation.

Circular Motion — Advanced Analysis

Derivation of Centripetal Acceleration: Consider a particle moving in a circle of radius $r$ with constant speed $v$. In a small time interval $\Delta t$, the particle moves from point P to Q. The velocity vectors $\vec{v_1}$ and $\vec{v_2}$ have the same magnitude $v$ but different directions. The change in velocity $\Delta\vec{v} = \vec{v_2} - \vec{v_1}$ is directed towards the centre of the circle.

Using the geometry of the triangle formed by the velocity vectors: $$\frac{|\Delta\vec{v}|}{v} = \frac{\Delta l}{r}$$ $$\Rightarrow |\Delta\vec{v}| = \frac{v\Delta l}{r}$$

Dividing by $\Delta t$ and taking limit $\Delta t \to 0$: $$a_c = \lim_{\Delta t \to 0} \frac{|\Delta\vec{v}|}{\Delta t} = \frac{v}{r} \lim_{\Delta t \to 0} \frac{\Delta l}{\Delta t} = \frac{v^2}{r}$$

Centrifugal Force (Pseudo Force): In a frame rotating with angular velocity $\omega$, a mass $m$ appears to experience an outward force $F_{cf} = m\omega^2 r = mv^2/r$. This is NOT a real force — it has no physical origin. It only appears because we are analysing from a non-inertial (accelerating) reference frame. Useful for understanding why passengers feel pushed outward in a turning vehicle.

Applications of Circular Motion:

1. Road banking: Reduces dependence on friction for providing centripetal force. National Highway equations use $\theta$ typically between 5°–15°.
2. Rounding a flat curve: Friction provides centripetal force: $f_s = \mu_s mg = mv^2/r \Rightarrow v_{max} = \sqrt{\mu_s rg}$
3. Conical pendulum: Used in road design and as a simple model for electron orbital angular momentum in some quantum pictures.
4. Loop-the-loop: Roller coasters, aircraft loops — minimum speed at top $v = \sqrt{rg}$ ensures the normal reaction doesn’t become zero before completing the loop.
5. Satellite motion (approximately circular): $F_c = G\frac{Mm}{r^2} = m\frac{v^2}{r} \Rightarrow v = \sqrt{\frac{GM}{r}}$

MDCAT Question Trends: Questions from circular motion commonly ask: (1) finding banking angle, (2) maximum speed for safe turn on flat road, (3) tension at different positions in vertical circular motion, (4) distinguishing centripetal vs centrifugal. PMDC/MDCAT papers from 2019–2024 show an average of 2–3 questions per paper on circular motion topics.

Key Points to Remember:

• $a_c$ always points towards the centre of the circular path
• Neither $a$ nor $v$ is zero in uniform circular motion — only $a_t = 0$
• The centripetal force can be any real force; it is not a new fundamental interaction
• In vertical motion, energy conservation supplements the force equations

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