Question:
A uniform bar of length 6 a and mass 8 m lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal plane with speed 2vand v respectively, strike the bar [as shown in the figure] and stick to the bar after collision. Denoting angular velocity (about the centre of mass), total energy and centre of mass velocity by $?$, E and $v_c$ respectively, we have after collision
A uniform bar of length 6 a and mass 8 m lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal plane with speed 2vand v respectively, strike the bar [as shown in the figure] and stick to the bar after collision. Denoting angular velocity (about the centre of mass), total energy and centre of mass velocity by $?$, E and $v_c$ respectively, we have after collision
Updated On: Jun 14, 2022
- $v_c=0$
- $?=\frac{3v}{5a}$
- $?=\frac{v}{5a}$
- $E=\frac{3}{5}mv^2$
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The Correct Option is D
Solution and Explanation
$P_i=0$
$\therefore P_f=0$ or $v_c=0$
$L_i=L_f or (2mv)a+(2a)(2a)=I?$ ....(i)
Here, $I=\frac{(8m)(6a)^2}{12}+m(2a)^2+(2m)(a)^2=30ma^2$
Substituting in E (i), we get
$?=\frac{v}{5a}$
Further, $E=\frac{1}{2}I?^2=\frac{1}{2}\times(30ma^2)\bigg(\frac{v}{5a}\bigg)^2=\frac{3mv^2}{5}$
$\therefore$ Correct options are (a), (c) and (d).
$\therefore P_f=0$ or $v_c=0$
$L_i=L_f or (2mv)a+(2a)(2a)=I?$ ....(i)
Here, $I=\frac{(8m)(6a)^2}{12}+m(2a)^2+(2m)(a)^2=30ma^2$
Substituting in E (i), we get
$?=\frac{v}{5a}$
Further, $E=\frac{1}{2}I?^2=\frac{1}{2}\times(30ma^2)\bigg(\frac{v}{5a}\bigg)^2=\frac{3mv^2}{5}$
$\therefore$ Correct options are (a), (c) and (d).
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Concepts Used:
System of Particles and Rotational Motion
- The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
- The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
- The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.