If L$\vec{L}$ and P$\vec{P}$ represent the angular momentum and linear momentum respectively of a particle of mass 'm$m$' having position vector r = a( i ω t + j ω t)$\vec{r} = a(\hat{i}\cos\omega t + \hat{j}\sin\omega t)$. The direction of force is

Related Formula

Velocity of a point on a rolling wheel at angular position θ$\theta$ from the lowest point:

Core Logic

At the highest point, θ = 180^°$\theta = 180^\circ$, so:

For a particle on the rim at the same horizontal level as the center, the angle from the lowest point is θ = 90^°$\theta = 90^\circ$.

Step 1: Compute Speed at Mid-Height

Substituting θ = 90^°$\theta = 90^\circ$ into our velocity relation: vmid = 2vcm (45^°) = 2 × 4 × 1√(2) = 4√(2) m/s$$v_{\text{mid}} = 2v_{\text{cm}} \sin(45^\circ) = 2 \times 4 \times \frac{1}{\sqrt{2}} = 4\sqrt{2}\text{ m/s}$$

Pattern Recognition

The contact point with the ground is the Instantaneous Center of Rotation (ICR). Distance to top point is 2R$2R$, distance to mid-level point is √(R²+R²) = √(2)R$\sqrt{R^2+R^2} = \sqrt{2}R$. Velocity scales linearly with distance from ICR.

Chapter Mix

Class 11 Physics: Rotational Motion

Related Formula

Conservation of Angular Momentum (since no external torque acts):

For a solid sphere, moment of inertia is:

Mass scales with volume: M ∝ R³$M \propto R^3$

Core Logic

When the radius reduces to R₂ = (R)/(2)$R_2 = \frac{R}{2}$, the mass scales cubically:

Now, compute the new moment of inertia I₂$I_2$:

Step 1: Compute Final Angular Velocity

Using conservation of angular momentum:

Hence, the value of x$x$ is 32.

Pattern Recognition

Since inertia of a solid sphere scales with M R²$M R^2$ and M ∝ R³$M \propto R^3$, the net moment of inertia scales with R⁵$R^5$. Shrinking the radius by half (1/2$1/2$) cuts down inertia by a factor of (1/2)⁵ = 1/32$(1/2)^5 = 1/32$. Velocity must scale up by 32 to conserve momentum.

Chapter Mix

Class 11 Physics: Rotational Motion

Related Formula

Fundamental mathematical definition of torque:

Rotational analogue of Newton's second law:

Linear momentum relations:

Core Logic

Let's check each expression sequentially:

• A. τ = I × L$\vec{\tau} = \vec{I} \times \vec{L}$ is dimensionally incorrect (Moment of inertia I$I$ is primarily treated as a tensor or scalar placeholder, not crossed directly like this).
• B. τ = d Ldt = (d)/(dt)( r × p)$\vec{\tau} = \frac{d\vec{L}}{dt} = \frac{d}{dt}(\vec{r} \times \vec{p})$ is fundamentally correct.
• C. τ = r × F = r × d pdt$\vec{\tau} = \vec{r} \times \vec{F} = \vec{r} \times \frac{d\vec{p}}{dt}$ is correct since F = d pdt$\vec{F} = \frac{d\vec{p}}{dt}$.
• D. τ = I α$\vec{\tau} = I\vec{\alpha}$ is the standard scalar component/fixed axis formulation.
• E. τ = r × F$\vec{\tau} = \vec{r} \times \vec{F}$ is the true physical vector definition.

Thus, statements B, C, D, and E are universally correct representations.

Pattern Recognition

Torque can be represented either through geometric structural parameters (position and force cross products) or via kinematic response properties (rate of change of angular momentum or product of rotational inertia and acceleration).

Chapter Mix

Class 11 Physics: System of Particles and Rotational Motion

Related Formula

Velocity of a rolling body from mechanical energy conservation:

Core Logic

Evaluate radius of gyration factor coefficients:

• For a circular ring: (k²)/(R²) = 1$\frac{k^2}{R^2} = 1$
• For a solid sphere: (k²)/(R²) = (2)/(5)$\frac{k^2}{R^2} = \frac{2}{5}$

Step 1: Compute Velocity Expressions

Step 2: Calculate the Velocity Ratio

Matching with the prompt expression format √((x)/(5))$\sqrt{\frac{x}{5}}$ reveals: x = 3.5$x = 3.5$

Rounding to the nearest integer yields 4.

Pattern Recognition

Objects with lower mass concentration near the center (lower (k²)/(R²)$\frac{k^2}{R^2}$ like the sphere) convert gravitational potential energy into translational kinetic energy more efficiently, rolling faster than hollow equivalents.

Chapter Mix

Class 11 Physics: System of Particles and Rotational Motion