A uniform disc of radius $R$ and mass $M$ is free to rotate only about its axis. A string is wrapped over its rim and a body of mass $m$ is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is
A uniform disc of radius $R$ and mass $M$ is free to rotate only about its axis. A string is wrapped over its rim and a body of mass $m$ is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is
- [JEE MAIN 2017]
- A
$\frac{{2mg}}{{2m + M}}$
- B
$\frac{{2Mg}}{{2m + M}}$
- C
$\frac{{2mg}}{{2M + m}}$
- D
$\frac{{2Mg}}{{2M + m}}$
Similar Questions
Write the conditions for equilibrium of a rigid body.
Write the conditions for equilibrium of a rigid body.
A ring is formed by joining two uniform semi circular rings $ABC$ and $ADC$. Mass of $ABC$ is thrice of that of $ADC$. If the ring is hinged to a fixed support ,at $A$, it can rotate freely in a vertical plane. Find the value of $tan\,\theta$, where $\theta$ is the angle made. by the line $AC$ with the vertical in equilibrium
A ring is formed by joining two uniform semi circular rings $ABC$ and $ADC$. Mass of $ABC$ is thrice of that of $ADC$. If the ring is hinged to a fixed support ,at $A$, it can rotate freely in a vertical plane. Find the value of $tan\,\theta$, where $\theta$ is the angle made. by the line $AC$ with the vertical in equilibrium
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]
Two men are carrying a uniform bar of length $L$, on their shoulders. The bar is held horizontally such that younger man gets $(1/4)^{th}$ load. Suppose the younger man is at the end of the bar, what is the distance of the other man from the end
Two men are carrying a uniform bar of length $L$, on their shoulders. The bar is held horizontally such that younger man gets $(1/4)^{th}$ load. Suppose the younger man is at the end of the bar, what is the distance of the other man from the end
An object of mass $8\,kg$ is hanging from one end of a uniform rod $CD$ of mass $2\,kg$ and length $1\,m$ pivoted at its end $C$ on a vertical wall as shown in figure. It is supported by a cable $A B$ such that the system is in equilibrium. The tension in the cable is $............\,N$ (Take $g=10\,m / s ^2$ )
An object of mass $8\,kg$ is hanging from one end of a uniform rod $CD$ of mass $2\,kg$ and length $1\,m$ pivoted at its end $C$ on a vertical wall as shown in figure. It is supported by a cable $A B$ such that the system is in equilibrium. The tension in the cable is $............\,N$ (Take $g=10\,m / s ^2$ )
- [JEE MAIN 2023]