A massless string is wrapped round a disc of mass $M$ and radius $R$. Another end is tied to a mass $m$ which is initially at height $h$ from ground level as shown in the fig. If the mass is released then its velocity while touching the ground level will be
A massless string is wrapped round a disc of mass $M$ and radius $R$. Another end is tied to a mass $m$ which is initially at height $h$ from ground level as shown in the fig. If the mass is released then its velocity while touching the ground level will be
- A
$\sqrt {2gh} $
- B
$\sqrt {2gh} \frac{M}{m}$
- C
$\sqrt {2gh\,m/M} $
- D
$\sqrt {4mgh/2m + M} $
Similar Questions
A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-intertial fiame of reference. The relationship between the force $\vec{F}_{\text {rot }}$ experienced by a particle of nass in moving on the rotating disc and the force $\vec{F}_{\text {in }}$ experienced by the particle in an inertial frame of reference is
$\vec{F}_{\text {rot }}=\vec{F}_{\text {in }}+2 m\left(\vec{v}_{\text {rot }} \times \vec{\omega}\right)+m(\vec{\omega} \times \vec{r}) \times \vec{\omega},$
where $\vec{v}_{\text {rot }}$ is the velocity of the particle in the rotating frame of reference and $\bar{r}$ is the position vector of the particle with respect to the centre of the disc.
Now consider a smooth slot along a diameter of a disc of radius $R$ rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the $x$-axis along the slot, the $y$-axis perpendicular to the slot and the $z$-axis along the rotation axis $(\vec{\omega}=\omega \hat{k})$. A sm a $1$ block of mass $m$ is gently placed in the slot at $\vec{r}=(R / 2) \hat{i}$ at $t=0$ and is constrained to move only along the slot.
(Image)
($1$) The distance $r$ of the block at time $t$ is
($A$) $\frac{R}{4}\left(e^{\omega t}+e^{-\omega t}\right)$ ($B$) $\frac{R}{2} \cos \omega t$ ($C$) $\frac{R}{4}\left(e^{2 \omega t}+e^{-2 \omega t}\right)$
($D$) $\frac{F}{2} \cos 2 \omega t$
($2$) The net reaction of the disc on the block is
($A$) $\frac{1}{2} m \omega^2 R\left(e^{2 \omega t}-e^{-2 \omega t}\right) \hat{j}+m g \hat{k}$
($B$) $\frac{1}{2} m \omega^2 R\left(e^{\omega t}-e^{-a t t}\right) j+m g k$
($C$) $-m \omega^2 R \cos \omega t \hat{j}-m g \hat{k}$
($D$) $m \omega^2 R \sin \omega t \hat{j}-m g \hat{k}$
Give the answer quetioin ($1$) ($2$)
A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-intertial fiame of reference. The relationship between the force $\vec{F}_{\text {rot }}$ experienced by a particle of nass in moving on the rotating disc and the force $\vec{F}_{\text {in }}$ experienced by the particle in an inertial frame of reference is
$\vec{F}_{\text {rot }}=\vec{F}_{\text {in }}+2 m\left(\vec{v}_{\text {rot }} \times \vec{\omega}\right)+m(\vec{\omega} \times \vec{r}) \times \vec{\omega},$
where $\vec{v}_{\text {rot }}$ is the velocity of the particle in the rotating frame of reference and $\bar{r}$ is the position vector of the particle with respect to the centre of the disc.
Now consider a smooth slot along a diameter of a disc of radius $R$ rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the $x$-axis along the slot, the $y$-axis perpendicular to the slot and the $z$-axis along the rotation axis $(\vec{\omega}=\omega \hat{k})$. A sm a $1$ block of mass $m$ is gently placed in the slot at $\vec{r}=(R / 2) \hat{i}$ at $t=0$ and is constrained to move only along the slot.
(Image)
($1$) The distance $r$ of the block at time $t$ is
($A$) $\frac{R}{4}\left(e^{\omega t}+e^{-\omega t}\right)$ ($B$) $\frac{R}{2} \cos \omega t$ ($C$) $\frac{R}{4}\left(e^{2 \omega t}+e^{-2 \omega t}\right)$
($D$) $\frac{F}{2} \cos 2 \omega t$
($2$) The net reaction of the disc on the block is
($A$) $\frac{1}{2} m \omega^2 R\left(e^{2 \omega t}-e^{-2 \omega t}\right) \hat{j}+m g \hat{k}$
($B$) $\frac{1}{2} m \omega^2 R\left(e^{\omega t}-e^{-a t t}\right) j+m g k$
($C$) $-m \omega^2 R \cos \omega t \hat{j}-m g \hat{k}$
($D$) $m \omega^2 R \sin \omega t \hat{j}-m g \hat{k}$
Give the answer quetioin ($1$) ($2$)
- [IIT 2016]
Write the conditions for equilibrium of a rigid body.
Write the conditions for equilibrium of a rigid body.
Consider the situation shown in the figure. Uniform rod of length $L$ can rotate freely about the hinge $A$ in vertical plane. Pulleys and string are light and frictionless. If therod remains horizontal at rest when the system is released then the mass of the rod is :
Consider the situation shown in the figure. Uniform rod of length $L$ can rotate freely about the hinge $A$ in vertical plane. Pulleys and string are light and frictionless. If therod remains horizontal at rest when the system is released then the mass of the rod is :
A uniform meter scale balances at the $40\,cm$ mark when weights of $10\,g$ and $20\,g$ are suspended from the $10\,cm$ and $20\,cm$ marks. The weight of the metre scale is ...... $g$
A uniform meter scale balances at the $40\,cm$ mark when weights of $10\,g$ and $20\,g$ are suspended from the $10\,cm$ and $20\,cm$ marks. The weight of the metre scale is ...... $g$
A uniform rod $AB$ of weight $100\, N$ rests on a rough peg at $C$ and $a$ force $F$ acts at $A$ as shown in figure. If $BC = CM$ and tana $= 4/3$. The minimum coefficient of friction at $C$ is
A uniform rod $AB$ of weight $100\, N$ rests on a rough peg at $C$ and $a$ force $F$ acts at $A$ as shown in figure. If $BC = CM$ and tana $= 4/3$. The minimum coefficient of friction at $C$ is