A flywheel can rotate in order to store kinetic energy. The flywheel is a uniform disk made of a material with a density $\rho $ and tensile strength $\sigma $ (measured in Pascals), a radius $r$ , and a thickness $h$ . The flywheel is rotating at the maximum possible angular velocity so that it does not break. Which of the following expression correctly gives the maximum kinetic energy per kilogram that can be stored in the flywheel ? Assume that $\alpha $ is a dimensionless constant
A flywheel can rotate in order to store kinetic energy. The flywheel is a uniform disk made of a material with a density $\rho $ and tensile strength $\sigma $ (measured in Pascals), a radius $r$ , and a thickness $h$ . The flywheel is rotating at the maximum possible angular velocity so that it does not break. Which of the following expression correctly gives the maximum kinetic energy per kilogram that can be stored in the flywheel ? Assume that $\alpha $ is a dimensionless constant
- A
$\alpha \sqrt {\frac{{\rho \sigma }}{r}} $
- B
$\alpha h\sqrt {\frac{{\rho \sigma }}{r}} $
- C
$\alpha \left( {\frac{h}{{{r^2}}}} \right)\left( {\frac{\sigma }{\rho }} \right)$
- D
$\frac{{\alpha \sigma }}{\rho }$
Similar Questions
A disc of mass $M$ and radius $R$ is rolling with angular speed $\omega $ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin $O$ is
A disc of mass $M$ and radius $R$ is rolling with angular speed $\omega $ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin $O$ is
A pendulum consists of a bob of mass $m=0.1 kg$ and a massless inextensible string of length $L=1.0 m$. It is suspended from a fixed point at height $H=0.9 m$ above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse $P=0.2 kg - m / s$ is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is $J kg - m ^2 / s$. The kinetic energy of the pendulum just after the lift-off is $K$ Joules.
($1$) The value of $J$ is. . . . . .
($2$) The value of $K$ is. . . . .
Give the answers of the questions ($1$) and ($2$)
A pendulum consists of a bob of mass $m=0.1 kg$ and a massless inextensible string of length $L=1.0 m$. It is suspended from a fixed point at height $H=0.9 m$ above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse $P=0.2 kg - m / s$ is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is $J kg - m ^2 / s$. The kinetic energy of the pendulum just after the lift-off is $K$ Joules.
($1$) The value of $J$ is. . . . . .
($2$) The value of $K$ is. . . . .
Give the answers of the questions ($1$) and ($2$)
- [IIT 2021]
Two thin circular discs of mass $m$ and $4 m$, having radii of $a$ and $2 a$, respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24} a$ through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point ' $O$ ' is $\vec{L}$ (see the figure). Which of the following statement($s$) is(are) true?
($A$) The center of mass of the assembly rotates about the $z$-axis with an angular speed of $\omega / 5$
($B$) The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81 m a^2 \omega$
($C$) The magnitude of angular momentum of the assembly about its center of mass is $17 \mathrm{ma}^2 \mathrm{\omega} / 2$
($D$) The magnitude of the $z$-component of $\vec{L}$ is $55 \mathrm{ma}^2 \omega$
Two thin circular discs of mass $m$ and $4 m$, having radii of $a$ and $2 a$, respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24} a$ through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point ' $O$ ' is $\vec{L}$ (see the figure). Which of the following statement($s$) is(are) true?
($A$) The center of mass of the assembly rotates about the $z$-axis with an angular speed of $\omega / 5$
($B$) The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81 m a^2 \omega$
($C$) The magnitude of angular momentum of the assembly about its center of mass is $17 \mathrm{ma}^2 \mathrm{\omega} / 2$
($D$) The magnitude of the $z$-component of $\vec{L}$ is $55 \mathrm{ma}^2 \omega$
- [IIT 2016]
Obtain $\tau = I\alpha $ from angular momentum of rigid body.
Obtain $\tau = I\alpha $ from angular momentum of rigid body.
A particle starts from the point $(0,8)$ metre and moves with uniform velocity of $\vec{v}=3 \hat{i} \,m / s$. What is the angular momentum of the particle after $5 \,s$ about origin is ........... $kg m ^2 / s$ (mass of particle is $1 \,kg$ )?
A particle starts from the point $(0,8)$ metre and moves with uniform velocity of $\vec{v}=3 \hat{i} \,m / s$. What is the angular momentum of the particle after $5 \,s$ about origin is ........... $kg m ^2 / s$ (mass of particle is $1 \,kg$ )?