Figure is an overhead view of a thin uniform rod of length 0.600mand mass Mrotating horizontally at 80.0rad/scounter clock-wise about an axis through its centre. A particle of mass M/3.00and travelling horizontally at speed 40.0m/shits the rod and sticks. The particle’s path is perpendicular to the rod at the instant of the hit, at a distance dfrom the rod’s centre.

(a) At what value of dare rod and particle stationary after the hit?

(b) In which direction do rod and particle rotate if dis greater than this value?

Short Answer

Expert verified
  1. Distance d is0.180m.
  2. Increasing the value of dwould cause the rod to rotate in clockwise direction.

Step by step solution

01

Step 1: Given

L=0.6mω=80rad/secv=40m/s

02

Determining the concept

Use law of conservation of angular momentum to find the distance. Also, use the formula for angular momentum in terms of mass, velocity and distance to find the angular momentum.According tothe conservation of momentum, momentum of a system is constant if no external forces are acting on the system.

Formula are as follow:

Li=LfL=mvr

Where,Lis angular momentum, mis mass, Vis velocity and r is radius.

03

Determining the distance d

(a)

From the law of conservation of angular momentum,

Li=Lfdmv+112ML2ω=0

Negative sign is taken for the clockwise rotation.

So,

d=ML2ω12mv

So,

d=M×0.62×8012×M3×40d=0.180m

Hence, the distance dis0.180m.

04

Determining the effect of increase in  d

(b)

If the lengthd is increased, system rotates clockwise. This is because, whend increases, termmvdin above equation would become more negative and hence would cause the clockwise rotation as per the sign conventions.

Hence, increasing the value of dwould cause the rod to rotate in clockwise direction.

Therefore, using the law of conservation of angular momentum and angular momentum formula, the required distance and the sense of rotation can be found if the direction is changed.

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