The question is based on Rotational Mechanics. Find the correct option (s)?

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1 Answer
Apr 6, 2017

Answer:

See below.

Explanation:

We have

#l=sqrt(24)a#
#l_g=l+1/5 l = 6/5l#
#vec r_g = (0,l_g/cosalpha,0)#

where #(u,v,w)# must be understood as

#u hat i + v hat j + w hat k#

Here

#alpha = arctan(1/sqrt(24))#
#vec omega = (0,cosalpha,sinalpha)omega#
#vec v_g = vec r_g xx vec omega = 17/2a^2m omega(0,cosalpha,sinalpha)#

then

#abs(vec v_g)=17/2a^2m omega#

and

(D) #Omega = abs(vec v_g)/abs(vec r_g) = 1/5 a omega#

or

#vec Omega = (0,0,1)Omega#

Now

#vec L_O = J_(omega) vec omega+J_(Omega) vec Omega#

with

#J_(omega)=(ma^2)/2+(4m(2a)^2)/2#

We have also

#vec L_g = J_(omega) vec omega# and

(B) #norm(vec L_g)=17/2a^2m omega#

and

#J_(Omega)=(ma^2)/4+ml_0^2+((4m)(2a)^2)/4+4m(2l_0)^2#

with #l_0 = l cos alpha#

so

(A) #<< vec L_O, hat k >> = J_(Omega)Omega+J_(omega)<< vec omega, hat k >> = J_(Omega)Omega+J_(omega)omega sin alpha#

and

#norm(vec L_O) = sqrt(<< vec L_O, hat k >>^2+(J_(omega)omega cosalpha)^2 )#

or

(C) #norm(vec L_O) = sqrt((J_(Omega)Omega)^2+2J_(Omega)J_(omega) Omega omega sinalpha+(J_(omega)omega)^2)#

The final numeric results are left to the reader as an exercise.