Rolling Motion — Revision Notes

⚡ 30-Second Revision

Pure Rolling Condition: — $v_{CM} = R\omega$
Total Kinetic Energy: — $K = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$
Acceleration on Incline: — $a_{CM} = \frac{g\sin\theta}{1 + \frac{I_{CM}}{MR^2}}$
Minimum Static Friction: — $\mu_{s,min} = \frac{\tan\theta}{1 + \frac{MR^2}{I_{CM}}}$
Work by Static Friction (Pure Rolling): — Zero
**Moment of Inertia ($I_{CM}$):**

* Ring/Hollow Cylinder: $MR^2$ * Disc/Solid Cylinder: $\frac{1}{2}MR^2$ * Solid Sphere: $\frac{2}{5}MR^2$ * Hollow Sphere: $\frac{2}{3}MR^2$

2-Minute Revision

Rolling motion is a combination of translation and rotation. For pure rolling, the point of contact is instantaneously at rest, leading to the crucial condition $v_{CM} = R\omega$. This means the total kinetic energy is the sum of translational ($\frac{1}{2}Mv_{CM}^2$) and rotational ($\frac{1}{2}I_{CM}\omega^2$) components.

When an object rolls down an inclined plane, its acceleration is $a_{CM} = \frac{g\sin\theta}{1 + \frac{I_{CM}}{MR^2}}$. Objects with smaller $I_{CM}/MR^2$ values (like a solid sphere) accelerate faster.

Static friction is essential for pure rolling on an incline, providing the necessary torque, but it does no work. Remember the moment of inertia values for common shapes: ring ($MR^2$), disc ($\frac{1}{2}MR^2$), solid sphere ($\frac{2}{5}MR^2$), hollow sphere ($\frac{2}{3}MR^2$).

These values are key to comparing the motion of different rolling bodies.

5-Minute Revision

Rolling motion is a composite motion involving both the linear movement of the center of mass (translation) and the spinning of the body about its center of mass (rotation). The defining characteristic of 'pure rolling' is the 'no-slip' condition, where the point of contact between the rolling body and the surface is instantaneously at rest.

This kinematic constraint is expressed as $v_{CM} = R\omega$, where $v_{CM}$ is the velocity of the center of mass, $R$ is the radius, and $omega$ is the angular velocity. This condition also implies that the point of contact acts as the instantaneous axis of rotation (IAOR), simplifying velocity calculations for other points on the body.

The total kinetic energy of a rolling body is the sum of its translational and rotational kinetic energies: $K_{total} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$. Using $v_{CM} = R\omega$, this can be rewritten as $K_{total} = \frac{1}{2}v_{CM}^2(M + \frac{I_{CM}}{R^2})$. The ratio of translational to rotational kinetic energy varies with the object's shape (moment of inertia).

When a body rolls down an inclined plane without slipping, static friction acts up the incline, providing the torque for rotation. Crucially, this static friction does no work because its point of application is momentarily at rest.

The acceleration of the center of mass is given by $a_{CM} = \frac{g\sin\theta}{1 + \frac{I_{CM}}{MR^2}}$. This formula shows that objects with smaller $I_{CM}/MR^2$ (e.g., solid sphere: $2/5$) will have greater acceleration than those with larger values (e.

g., ring: $1$). The minimum coefficient of static friction required for pure rolling is $\mu_{s,min} = \frac{\tan\theta}{1 + \frac{MR^2}{I_{CM}}}$. Remember the moments of inertia for common shapes: Ring ($MR^2$), Disc ($\frac{1}{2}MR^2$), Solid Sphere ($\frac{2}{5}MR^2$), Hollow Sphere ($\frac{2}{3}MR^2$).

Example: A solid cylinder rolls down an incline. $I_{CM} = \frac{1}{2}MR^2$. Its acceleration is $a_{CM} = \frac{g\sin\theta}{1 + \frac{1/2 MR^2}{MR^2}} = \frac{g\sin\theta}{1 + 1/2} = \frac{2}{3}g\sin\theta$. Its total KE is $K = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}(\frac{1}{2}MR^2)(\frac{v_{CM}}{R})^2 = \frac{1}{2}Mv_{CM}^2 + \frac{1}{4}Mv_{CM}^2 = \frac{3}{4}Mv_{CM}^2$. The ratio $K_{trans}:K_{rot} = (\frac{1}{2}Mv_{CM}^2) : (\frac{1}{4}Mv_{CM}^2) = 2:1$.

Prelims Revision Notes

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Definition: — Rolling motion is a combination of translational motion of the center of mass and rotational motion about the center of mass.
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Pure Rolling (No Slipping):

* Condition: $v_{CM} = R\omega$. The instantaneous velocity of the point of contact with the surface is zero. * Instantaneous Axis of Rotation (IAOR): The point of contact acts as the IAOR. Velocity of any point P is $v_P = r_{IAOR}\omega$, where $r_{IAOR}$ is the distance from P to the contact point. * Velocity at top-most point: $2v_{CM}$. * Velocity at bottom-most point: $0$.

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Kinetic Energy of Rolling:

* Total Kinetic Energy: $K_{total} = K_{translational} + K_{rotational} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$. * Using $v_{CM} = R\omega$, $K_{total} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}(\frac{v_{CM}}{R})^2 = \frac{1}{2}v_{CM}^2(M + \frac{I_{CM}}{R^2})$. * Alternatively, using IAOR: $K_{total} = \frac{1}{2}I_{IAOR}\omega^2 = \frac{1}{2}(I_{CM} + MR^2)\omega^2$.

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Moment of Inertia ($I_{CM}$) for Common Shapes: — (Crucial for NEET)

* Ring / Hollow Cylinder: $MR^2$ * Disc / Solid Cylinder: $\frac{1}{2}MR^2$ * Solid Sphere: $\frac{2}{5}MR^2$ * Hollow Sphere: $\frac{2}{3}MR^2$

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Rolling Down an Inclined Plane:

* Forces: $Mg\sin\theta$ (down incline), $f_s$ (static friction, up incline), $N$ (normal force). * Acceleration of CM: $a_{CM} = \frac{g\sin\theta}{1 + \frac{I_{CM}}{MR^2}}$. * Time to reach bottom: $t = \sqrt{\frac{2h(1 + I_{CM}/MR^2)}{g\sin^2\theta}}$.

(From $h = \frac{1}{2}a_{CM}t^2\sin\theta$ and $s = h/\sin\theta$) * Velocity at bottom: $v_{CM} = \sqrt{\frac{2gh}{1 + I_{CM}/MR^2}}$. (From energy conservation $Mgh = K_{total}$) * Order of arrival: Objects with smaller $I_{CM}/MR^2$ (e.

g., solid sphere) reach the bottom first, followed by disc, then hollow sphere, then ring.

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Role of Static Friction:

* Provides the necessary torque for angular acceleration. * Does NO WORK in pure rolling, as the point of application is instantaneously at rest. * Minimum coefficient of static friction for pure rolling: $\mu_{s,min} = \frac{\tan\theta}{1 + \frac{MR^2}{I_{CM}}}$. If $\mu_s < \mu_{s,min}$, slipping occurs.

Vyyuha Quick Recall

To remember the order of objects rolling down an incline (fastest to slowest): Solid Sphere, Solid Cylinder, Hollow Sphere, Hollow Cylinder/Ring.

Think: Super Speedy Cars Have Smooth Heels. (Solid Sphere, Solid Cylinder, Hollow Sphere, Hollow Cylinder/Ring). This mnemonic helps recall the order based on their $I_{CM}/MR^2$ values (smallest to largest).