Question

1. A thin, uniform bar of length L and mass M is supported by a frictionless horizontal surface. A ball of mass m travels to the right with speed v and strikes the bar. (top view) a. Assume the ball strikes the bar a distance d from the center of the bar and sticks to the bar. Find the velocity of the center of mass of the bar-ball system and them angular speed of the system. b. Assume the ball strikes the bar a distance L/4 from the center of the bar. This time, the collision is elastic and the ball comes to rest. Find the radio m/M.

Answer 1

Given . Mass of bar = M. mass of ball am velocity of ball = 3 Акт () mot v → Ball 'd - - strikes at a distance from centre of bar. After striking ball sticks to .. bar Since there is no external force acting in the system ( ball of rod)

So, linear momentum is conserved (Pi =P ) combined let vf be the velocity of is bar and ball mv + MV = (m+ M) VA V→ initial velocity of 1 bar =0 mu to to = (m+1) VA s mu velocity of centre of muss of bas-ball

Angular velocity of system Since there is no external torque acting in the system So, Angular momentum (2) is conserved Paking angular momentum about "os centre of mass of i bar before collision} Ž: mud - Isystem w - ) co y final angular O speed of bar. Isystem » Moment of - Inertia of bar + ball about o

moment of inertia of bar about its centre of mass = Me moment of inertiá of: ball about 'o' = mer? Lsystem el present euro from equation (i) mrd = Icm to) mud Elmo her more you - mud = [142 + 12 md) L 12 w = 112 muda L ML²+ 12 md2 - Angular velocity of system /2md C

PART (6) Collision is elastie -) » 1. of Energy is conserved y Ball comes to rest & This time ball doesn't I stick to bas ? 1 torce Since mo any external force is acting. - - linear momentum is conserved » mv = mvf, t. Vfz tyxИ v initial velocity of bas =ó final velocity Vfi Ufa Vfr of ball to final velocity of bary of

=) muto= o+N Vf As external torque es absent, So, angular momentum O is conserved. Li a lf n 7) mvby) Ibar w s Angular momentum about centre of mass - of bar ? 12 mulle meew To a 3 m do sad /s] - 6 3 my ML

As colliscon is elastic Energy is conserved - £mv² = £mvh + 2 MV 2 a) {mv? 0+ Vz? a) į mi a { MV2 + Escolese - Now putting value of Vfe B & w from equation (a) - ļ (6) 7) 1 mv² = + M m

을뤄씀 루 서 NK ON K NS 는 뚱 일 은 시루 서 두 77 |