Class 11PhysicsSystem of Particles and Rotational MotionMix Example - System of Particles and Rotational Motion
Difficult$A$ rod of mass $m$ and length $L$,pivoted at one of its ends,is hanging vertically. $A$ bullet of the same mass moving at speed $v$ strikes the rod horizontally at a distance $x$ from its pivoted end and gets embedded in it. The combined system now rotates with angular speed $\omega$ about the pivot. The maximum angular speed $\omega_M$ is achieved for $x=x_M$. Then
$(A)$ $\omega=\frac{3 v x}{ L ^2+3 x^2}$
$(B)$ $\omega=\frac{12 v x}{L^2+12 x^2}$
$(C)$ $x_M=\frac{L}{\sqrt{3}}$
$(D)$ $\omega_M=\frac{v}{2 L} \sqrt{3}$
$(A)$ $\omega=\frac{3 v x}{ L ^2+3 x^2}$
$(B)$ $\omega=\frac{12 v x}{L^2+12 x^2}$
$(C)$ $x_M=\frac{L}{\sqrt{3}}$
$(D)$ $\omega_M=\frac{v}{2 L} \sqrt{3}$
- A$A, B, C$
- B$A, B, D$
- C$A, C, D$
- D$A, C$
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Similar Questions
$A$ uniform sphere of mass $m$ and radius $R$ is placed on a rough horizontal surface. The sphere is struck horizontally at a height $h$ from the floor. Match the following:
$(a)$ $h = \frac{R}{2}$ $(i)$ Sphere rolls without slipping with a constant velocity and no loss of energy. $(b)$ $h = R$ $(ii)$ Sphere spins clockwise,loses energy by friction. $(c)$ $h = \frac{3R}{2}$ $(iii)$ Sphere spins anti-clockwise,loses energy by friction. $(d)$ $h = \frac{7R}{5}$ $(iv)$ Sphere has only a translational motion,loses energy by friction.
| $(a)$ $h = \frac{R}{2}$ | $(i)$ Sphere rolls without slipping with a constant velocity and no loss of energy. |
| $(b)$ $h = R$ | $(ii)$ Sphere spins clockwise,loses energy by friction. |
| $(c)$ $h = \frac{3R}{2}$ | $(iii)$ Sphere spins anti-clockwise,loses energy by friction. |
| $(d)$ $h = \frac{7R}{5}$ | $(iv)$ Sphere has only a translational motion,loses energy by friction. |
Medium
View Solution$A$ non-uniform cylinder of mass $m$,length $l$,and radius $r$ has its center of mass at a distance $l/4$ from the geometric center,lying on the axis of the cylinder. The cylinder is kept in a liquid of uniform density $\rho$. The moment of inertia of the cylinder about its center of mass is $I$. The angular acceleration of the cylinder just after it is released from the horizontal position shown in the figure is:
Difficult
View Solution$A$ body with a moment of inertia of $3 \ kg \cdot m^2$ rotating with an angular speed of $2 \ rad/s$ has the same kinetic energy as a mass of $12 \ kg$ moving with a speed of ......... $m/s$.
Medium
View Solution$A$ rod of length $1\,m$ is standing vertically. When its upper end is released and it falls such that the lower end touches the ground without slipping,the speed of the upper end when it hits the ground is:
Medium
View SolutionThree solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side length $d$. The ratio $I_{0} / I_{A}$ of the moment of inertia $I_{0}$ of the system about an axis passing through the centroid and perpendicular to the plane of the triangle,to the moment of inertia $I_{A}$ about an axis passing through the center of any one of the spheres and perpendicular to the plane of the triangle,is:
DifficultJEE MAIN 2020
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