Oscillations and Rotational Dynamics Question

[hairyshan]

Hello Sir,
I have a question regarding springs
Q. Two identical objects of mass m are placed at either end of a spring of spring constant k and the whole system is placed on a horizontal frictionless surface. At what angular frequency does the system oscillate in terms of m and k?
The other question regarding rotational dynamics:
Q. Two discs are mounted on thin, lightweight rods orientated through their centres and normal to the discs. These axles are constrained to be vertical at all times, and the discs can pivot frictionlessly on the rods. The discs have identical thicknesses and are made of the same material, but have differing radii r1 and r2. The discs are given angular velocity w1 and w2, respectively and are brought into contact at their edges. After the discs interact, it is seen that they both come to a halt (they are held together in contact such that they rub against each other - so this obeys conservation of angular momentum). Show that the following condition: w1. r1^3 = w2.r2^3 hold
Thanks,

[hairyshan]

Also i wanted to mention that for the second question, I keep getting w1r1^4 = w2r2^4 instead of the desired result.

[joe]

Okay. In that case I can answer.

Q. Two identical objects of mass m are placed at either end of a spring of spring constant k and the whole system is placed on a horizontal frictionless surface. At what angular frequency does the system oscillate in terms of m and k?

No external forces so centre of mass stationary. From symmetry, the centre of the spring is stationary and the masses move in antisymmetry.
Let the displacement of RH mass from rest be x and LH mass be –x.
So spring is stretched 2x, so |tension| is 2kx. So for each mass we have
m.d^2/dt^2 = –2kx.
so omega = sqrt(2k/m)

You can think of cutting the spring in half (so spring with constant 2k) and fixing one end and m on the other.

Joe

[joe]

Q. Two discs are mounted on thin, lightweight rods orientated through their centres and normal to the discs. These axles are constrained to be vertical at all times, and the discs can pivot frictionlessly on the rods. The discs have identical thicknesses and are made of the same material, but have differing radii r1 and r2. The discs are given angular velocity w1 and w2, respectively and are brought into contact at their edges. After the discs interact, it is seen that they both come to a halt (they are held together in contact such that they rub against each other - so this obeys conservation of angular momentum). Show that the following condition: w1. r1^3 = w2.r2^3 hold

No external torques so conservation of angular momentum so
initial angular momentum = I2w2 + I1w1 = 0 = final angular momentum.
But I is mr^2/2 and m is also proportional to r^2 so I get an answer much like yours (w1r1^4 = – w2r2^4, but note the minus sign), not that of the question.

Joe

[hairyshan]

ok I understand. Thank you very much for the quick reply.