A \"diatomic\" system. Two masses m1 and m2 are connected by a massless spring w

Question

A "diatomic" system. Two masses m1 and m2 are connected by a massless spring with spring constant k. If the masses arc displaced from their equilibrium positions, we wish to find the angular frequency to of the resulting oscillations. Let the masses be displaced by x1 and x2 as shown in the figure above (where a positive displacement of each mass stretches the spring). What is the force on mass m1? (Make sure that sign is correct, so that the direction of the force is consistent with the direction of x1.) What is the force on mass m2? (Again, make sure that the direction of the force is consistent with the direction of x2.) Use Newton's law for each mass to obtain separate expressions for d2x1/dt2 and d2x2/dt2. (Note: each 2nd derivative should be proportional to x1+x2.) Add the resulting equations from part (b) to obtain an equation for simple harmonic motion for the total separation (x1+x2) between the two masses. This equation should be in the form d2(x1+ x2)/dt2 = -omega2(x1+x2). What is the angular frequency omega of the resulting oscillations in terms of m1, m2, and k? Show that the angular frequency omega is the same as the frequency of a single particle of mass M = m1m2/(m1+m2) attached to one end of a spring with a spring constant k and the other end fixed. (M is called the "reduced mass" of the system.) If k = 500 N/m. m1 = 0.03 kg. and m2 = 0.05 kg. what are the reduced mass M and the oscillation frequency to?

Explanation / Answer

a)
i) The force of a spring in magnitude is equal to k * stretch of the spring.
In this problem the stretch of the spring is x1+x2
Thus the force for mass 1 is -k(x1+x2) (negative since the spring will pull m1 opposite to x1)

ii) Similarly for mass 2 we will find -k(x1+x2)

b) Since we know F=ma we can write down the answer

m1d2x1/dt2=-k(x1+x2) and m2d2x2/dt2=-k(x1+x2)

c) just divide both sides by their m and add together to get d2(x1+x2)/dt2=-(k/m1+k/m2)(x1+x2)

or combining denominators: d2(x1+x2)/dt2=-k(m2+m1)/(m2m1)(x1+x2)

since for simple harmonic motion d2(x1+x2)/dt2=-2(x1+x2)

Thus =k(m2+m1)/(m2m1)=163.3

d) since for a simple spring system =k/m we can see that is the same as a particle just with the reduced mass.

M=0.01875

and =

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.