Circular Motion — Revision Notes

⚡ 30-Second Revision

Angular Velocity: — $\omega = \frac{d\theta}{dt} = \frac{v}{r}$ (rad/s)
Angular Acceleration: — $\alpha = \frac{d\omega}{dt} = \frac{a_t}{r}$ (rad/s$^2$)
Centripetal Acceleration: — $a_c = \frac{v^2}{r} = r\omega^2$ (towards center)
Tangential Acceleration: — $a_t = \frac{dv}{dt} = r\alpha$ (tangent to path)
Total Acceleration (NUCM): — $a = \sqrt{a_c^2 + a_t^2}$
Centripetal Force: — $F_c = ma_c = \frac{mv^2}{r} = mr\omega^2$ (towards center)
Banking of Roads: — $\tan\theta = \frac{v^2}{rg}$
Vertical Circle (String): — $v_{min,top} = \sqrt{rg}$, $v_{min,bottom} = \sqrt{5rg}$
Work by Centripetal Force: — $W = 0$ (always perpendicular to displacement)

2-Minute Revision

Circular motion involves an object moving along a circular path. It's crucial to distinguish between uniform (constant speed) and non-uniform (changing speed) motion. Even in uniform circular motion, velocity changes direction, leading to centripetal acceleration ($a_c = v^2/r = r\omega^2$) directed towards the center.

This acceleration is caused by a centripetal force ($F_c = mv^2/r = mr\omega^2$), which is always provided by an existing physical force (e.g., tension, friction, gravity). The work done by centripetal force is always zero as it's perpendicular to displacement.

\n\nIn non-uniform circular motion, speed also changes, introducing a tangential acceleration ($a_t = dv/dt = r\alpha$) along the path. The total acceleration is the vector sum of $a_c$ and $a_t$.

Key applications include banking of roads (angle $\tan\theta = v^2/rg$) and motion in a vertical circle. For a vertical circle with a string, the minimum speed at the top to complete the loop is $\sqrt{rg}$, and at the bottom is $\sqrt{5rg}$.

Remember to convert units and draw free-body diagrams for complex problems.

5-Minute Revision

Circular motion is the movement of an object along a circular path. The core idea is that even if the speed is constant (Uniform Circular Motion, UCM), the direction of velocity continuously changes, implying an acceleration.

This is the **centripetal acceleration ($a_c$)**, always directed towards the center, with magnitude $a_c = v^2/r = r\omega^2$. Here, $v$ is linear speed, $r$ is radius, and $\omega$ is angular speed.

This acceleration requires a **centripetal force ($F_c$)**, given by $F_c = ma_c = mv^2/r = mr\omega^2$. This force is *not* a new fundamental force; it's provided by other forces like tension, friction, or gravity.

Crucially, the work done by centripetal force is always zero because it's perpendicular to the instantaneous displacement. \n\nIn Non-Uniform Circular Motion (NUCM), the object's speed also changes.

This introduces a **tangential acceleration ($a_t$)** along the path, with magnitude $a_t = dv/dt = r\alpha$, where $\alpha$ is angular acceleration. The total acceleration in NUCM is the vector sum $a = \sqrt{a_c^2 + a_t^2}$.

\n\nKey Applications: \n1. Banking of Roads: To safely negotiate turns at higher speeds, roads are banked. The ideal banking angle $\theta$ is given by $\tan\theta = v^2/(rg)$. This formula helps calculate the speed for which no friction is required.

\n2. Conical Pendulum: A mass swinging in a horizontal circle, with the string making a constant angle with the vertical. The horizontal component of tension provides the centripetal force. \n3. Motion in a Vertical Circle: This is a classic NUCM example.

Speed and tension vary. For a string, the minimum speed at the highest point to complete the circle is $v_{top} = \sqrt{rg}$. Using conservation of energy, the minimum speed at the lowest point is $v_{bottom} = \sqrt{5rg}$.

Tension at the bottom is $T_{bottom} = mg + mv^2/r$, and at the top is $T_{top} = mv^2/r - mg$. \n\nExample: A $0.1,\text{kg}$ ball is whirled in a horizontal circle of $0.5,\text{m}$ radius at $2,\text{m/s}$.

The centripetal force (tension) is $F_c = (0.1,\text{kg})(2,\text{m/s})^2 / (0.5,\text{m}) = 0.4 / 0.5 = 0.8,\text{N}$. \n\nRemember to convert units (e.g., km/h to m/s, rpm to rad/s) and always draw free-body diagrams for force analysis.

Prelims Revision Notes

Circular Motion: NEET Revision Notes

1. Basic Definitions & Relations:

Angular Displacement ($\theta$): — Angle swept by radius vector. Unit: radian (rad).
Angular Velocity ($\omega$): — Rate of change of angular displacement. $\omega = \frac{d\theta}{dt}$. Unit: rad/s. Relation to linear speed: $v = r\omega$.
Angular Acceleration ($\alpha$): — Rate of change of angular velocity. $\alpha = \frac{d\omega}{dt}$. Unit: rad/s$^2$. Relation to tangential acceleration: $a_t = r\alpha$.
Period (T): — Time for one complete revolution. $T = \frac{2\pi}{\omega}$.
Frequency (f): — Number of revolutions per second. $f = \frac{1}{T} = \frac{\omega}{2\pi}$.

2. Uniform Circular Motion (UCM):

Speed: — Constant.
Velocity: — Magnitude constant, direction continuously changes.
Centripetal Acceleration ($a_c$): — Always present, directed towards the center. $a_c = \frac{v^2}{r} = r\omega^2$. Its magnitude is constant.
Tangential Acceleration ($a_t$): — Zero.
Centripetal Force ($F_c$): — Always present, directed towards the center. $F_c = ma_c = \frac{mv^2}{r} = mr\omega^2$. Provided by external forces (e.g., tension, friction, gravity).
Work Done by Centripetal Force: — Zero, as $F_c \perp \vec{v}$ (or displacement).

3. Non-Uniform Circular Motion (NUCM):

Speed: — Varies (changes).
Velocity: — Both magnitude and direction change.
Centripetal Acceleration ($a_c$): — Present, directed towards the center. Magnitude $a_c = \frac{v^2}{r}$ varies as $v$ changes.
Tangential Acceleration ($a_t$): — Present, directed tangent to the path. $a_t = \frac{dv}{dt} = r\alpha$. Responsible for change in speed.
Total Acceleration ($a$): — Vector sum of $a_c$ and $a_t$. $a = \sqrt{a_c^2 + a_t^2}$.

4. Applications:

Banking of Roads: — For safe turning at speed $v$ on a road of radius $r$, the ideal banking angle $\theta$ is given by $\tan\theta = \frac{v^2}{rg}$. If friction is considered, the maximum safe speed is more complex.
Conical Pendulum: — A mass $m$ on a string of length $L$ moving in a horizontal circle. Tension $T$ has components: $T\cos\theta = mg$ and $T\sin\theta = \frac{mv^2}{r}$. Period $T_{period} = 2\pi \sqrt{\frac{L\cos\theta}{g}}$.
Motion in a Vertical Circle: — (Non-uniform motion)

* String: Minimum speed at top ($v_{top}$) to complete circle: $v_{top} = \sqrt{rg}$. Minimum speed at bottom ($v_{bottom}$) to complete circle: $v_{bottom} = \sqrt{5rg}$. * Tension at bottom: $T_L = mg + \frac{mv_L^2}{r}$. * Tension at top: $T_H = \frac{mv_H^2}{r} - mg$. * Difference in tension: $T_L - T_H = 2mg + \frac{m}{r}(v_L^2 - v_H^2)$. * Rod: Minimum speed at top can be $0$ (rod can push). Minimum speed at bottom $v_{bottom} = \sqrt{4rg}$.

5. Common Traps:

Confusing centripetal and centrifugal forces (centrifugal is fictitious).
Assuming constant speed implies zero acceleration in circular motion.
Incorrect unit conversions.
Errors in force diagrams for vertical circles or banking.

Vyyuha Quick Recall

Can My Velocity Always Change For Radius?

Centripetal Motion: Circular Motion
Velocity: Direction always changes (even if speed is constant)
Always Change: Implies Centripetal Acceleration ($a_c = v^2/r$)
For Radius: This acceleration needs a Force (Centripetal Force, $F_c = mv^2/r$) directed towards the Radius (center).