A pendulum is formed by pivoting a disc. What | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) For a physical pendulum,the time period is given by $T = 2\pi \sqrt{\frac{I}{mg\ell}}$,where $I$ is the moment of inertia about the pivot point an

Vedclass · Accepted Answer

$\frac{R}{\sqrt{2}}$

Vedclass · Answer

(B) For a physical pendulum,the time period is given by $T = 2\pi \sqrt{\frac{I}{mg\ell}}$,where $I$ is the moment of inertia about the pivot point and $\ell$ is the distance from the center of mass $(COM)$ to the pivot.Using the parallel axis theorem,$I = I_{COM} + m\ell^2 = \frac{1}{2}mR^2 + m\ell^2$.Substituting this into the formula: $T = 2\pi \sqrt{\frac{\frac{1}{2}mR^2 + m\ell^2}{mg\ell}} = 2\pi \sqrt{\frac{R^2 + 2\ell^2}{2g\ell}}$.For $T$ to be minimum,the term $f(\ell) = \frac{R^2 + 2\ell^2}{\ell} = \frac{R^2}{\ell} + 2\ell$ must be minimum.Differentiating with respect to $\ell$ and setting to zero: $\frac{df}{d\ell} = -\frac{R^2}{\ell^2} + 2 = 0$.Solving for $\ell$: $2\ell^2 = R^2 \implies \ell = \frac{R}{\sqrt{2}}$.

$A$ pendulum is formed by pivoting a disc. What distance from the center of mass should it be pivoted for the minimum time period while performing $SHM$?

Similar Questions

$A$ particle of mass $m$ is located in a one-dimensional potential field where the potential energy is given by: $V(x) = A(1 - \cos px)$, where $A$ and $p$ are constants. The period of small oscillations of the particle is

$A$ cylindrical block of mass $M$ and area of cross-section $A$ is floating in a liquid of density $\rho$ with its axis vertical. When depressed a little and released, the block starts oscillating. The period of oscillation is . . . . . . . . . . .

$A$ uniform rod of mass $m$ and length $l$ is suspended about its end. The time period of small angular oscillations is ..........

$A$ ring is hung on a nail. It can oscillate,without slipping or sliding $(i)$ in its plane with a time period $T_{1}$ and,$(ii)$ back and forth in a direction perpendicular to its plane,with a period $T_{2}$. The ratio $\frac{T_{1}}{T_{2}}$ will be

What factors determine the natural frequency of oscillation of a body?

Vedclass Test Series

Exam Paper Generator

Online Exam Module