A rod of length l and mass m lies on a fixed | vedclass

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

Question

(C) Let $a_{cm}$ be the acceleration of the centre of mass of the rod and $\alpha$ be the angular acceleration about the centre of mass. The accelerat

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(C) Let $a_{cm}$ be the acceleration of the centre of mass of the rod and $\alpha$ be the angular acceleration about the centre of mass. The acceleration of the end of the rod where the string is attached is $a = a_{cm} + \alpha (l/2)$.From the equation of motion for the block $m_1$: $m_1 g - T = m_1 a$.For the rod,the force equation is $T = m a_{cm}$ and the torque equation about the centre of mass is $T(l/2) = I \alpha = (ml^2/12) \alpha$,which gives $T = (m l / 6) \alpha$,so $a_{cm} = (l/6) \alpha$.Substituting $\alpha = 6 a_{cm} / l$ into the expression for $a$: $a = a_{cm} + (l/2)(6 a_{cm} / l) = a_{cm} + 3 a_{cm} = 4 a_{cm}$.Substituting $T = m a_{cm}$ and $a = 4 a_{cm}$ into the block's equation: $m_1 g - m a_{cm} = m_1 (4 a_{cm}) \implies m_1 g = (4 m_1 + m) a_{cm}$.Thus,$a_{cm} = \frac{m_1}{4 m_1 + m} g = \frac{g}{4 + (m/m_1)}$.To maximize $a_{cm}$,we need to minimize the denominator $4 + (m/m_1)$. As $m_1 	o \infty$,$m/m_1 	o 0$,so $a_{cm, max} = g/4$. Therefore,$n = 4$.

Similar Questions

$A$ mass $m$ is supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder,with what acceleration will the mass fall on release?

$A$ ring has inner radius $R_1$ and outer radius $R_2$. It rolls without slipping with a constant angular velocity $\omega$. What is the ratio of the forces $F_1/F_2$ experienced by two particles located on the inner and outer surfaces of the ring?

$A$ rigid body rotates about a fixed axis with variable angular velocity equal to $\alpha - \beta t$,where $t$ is time and $\alpha, \beta$ are constants. Find the angle (in radians) through which it rotates before it stops.

$A$ uniform rod of mass $m$ and length $l$ rotates in a horizontal plane with an angular velocity $\omega$ about a vertical axis passing through one end. The tension in the rod at a distance $x$ from the axis is

$A$ rigid rod of length $\ell$ is sliding. At some instant,the position of the rod is as shown in the figure. End $A$ has a constant velocity $v_0$ along the $x$-axis. At $t = 0$,the rod is vertical with end $B$ at $y = \ell$. Which of the following statements is correct?

Vedclass Test Series

Exam Paper Generator

Online Exam Module