Circular Motion – Centripetal Force, Banking of Roads & Key Formulas | Entrance Exam Physics Notes

CIRCULAR MOTION — SHORT NOTES (Exam Focus)

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1. Definition

Circular motion occurs when an object moves along a circular path with radius r.

Two types:

1. Uniform Circular Motion (UCM) – Speed constant

2. Non-uniform Circular Motion – Speed changes

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2. Angular Quantities

Angular displacement (θ)

Measured in radians.

Angular velocity (ω)

$$\omega =\frac{d\theta }{d\mathrm{t}}$$

Angular acceleration (α)

$$\alpha =\frac{d\omega }{dt}$$

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3. Relations Between Linear & Angular Quantities

$$v = r\omega$$
$$a_t=r\alpha$$

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4. Centripetal Acceleration (ac)

Acceleration directed towards the center in circular motion.

$$a_c=\frac{v^2}{r}=r{\omega }^2$$

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5. Centripetal Force (Fc)

Force required to keep an object moving in a circle.

$$F_c=m{a}_c=\frac{m{v}^2}{r}=mr{\omega }^2$$

Important: Centripetal force is not a separate force.
It is provided by tension, gravity, friction, normal reaction, etc.

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6. Centrifugal Force (Pseudo Force)

In a rotating (non-inertial) frame, a pseudo force acts outward:

$$F_{cf}=\frac{m{v}^2}{r}$$

Not a real force — only appears in rotating frame.

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7. Time Period & Frequency

$$T = \frac{2\pi r}{v}$$ $$f = \frac{1}{T}$$ $$\omega = 2\pi f$$

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8. Banking of Roads

Used to avoid slipping/skidding when vehicles turn.

Bank angle (θ) for no friction:

$$\tan \theta =\frac{v^2}{r\mathrm{g}}$$

Safe speed (no skidding):

$$v=\sqrt{rg\tan \theta }$$

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9. Motion in a Vertical Circle

Important for exam numericals.

Minimum speed at top for completing loop:

$$v_{\text{top}}=\sqrt{gr}$$

Minimum speed at bottom:

$$v_{\text{bottom}}=\sqrt{5gr}$$

Tensions:

$$T_{\text{top}} = \frac{mv^2}{r} - mg$$
$$T_{\text{bottom}} = \frac{mv^2}{r} + mg$$

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10. Conical Pendulum

A mass rotates making an angle θ with vertical.

Key formulas:

$$T \cos\theta = mg$$
$$T \sin\theta = \frac{mv^2}{r}$$ $$\omega =\sqrt{\frac{g}{l\cos \theta }}$$

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11. Rolling Motion (basic intro)

For a rolling wheel:

$$v=r\omega$$

Point of contact with ground has zero velocity.

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12. Common Exam Results

1. Friction acting as centripetal force

Car turning on horizontal road:

$$f=\frac{m{v}^2}{\mathrm{r}}$$

 Condition for safe turning:

$$\frac{v^2}{r}\le \mu g$$

2. Tension in rotating string

$$T=\frac{m{v}^2}{\mathrm{r}}$$

3. Rotating body on turntable

Object slips if required centripetal force > friction.

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13. Most Expected Numericals

1. Safe speed on curved road (banked/unbanked)

2. Car turning on rough vs smooth road

3. Vertical circle minimum speed + tensions

4. Mass whirling in horizontal circle

5. Conical pendulum period & tension

6. Centripetal force balance problems

7. Loop-the-loop problems (rollercoaster physics)