Two springs in parallel Consider two massless springs connected in parallel. Spr

Question

Two springs in parallel Consider two massless springs connected in parallel. Springs 1 and 2 have spring constants k_1 and k_2 and are connected via a thin, vertical rod. A constant force of magnitude F is being exerted on the rod. The rod remains perpendicular to the direction of the applied force, so that the springs are extended by the same amount. This system of two springs is equivalent to a single spring, of spring constant k. Part A Find the effective spring constant k of the two-spring system. Give your answer for the effective spring constant in terms of k_1 and k_2. k = --- Three springs in parallel Now consider three springs connected in parallel as shown. The spring constants of springs 1, 2, and 3 are k_1, k_2, and k_3. The springs are connected by a vertical rod, and a force of magnitude F is being exerted to the right. Part B Find the effective spring constant k' of the three-spring system. Give your answer in terms of k_1, k_2, and k_3. k' =

Explanation / Answer

When you have springs in series, the effective spring constant is given by 1/k = 1/k1 + 1/k2 And compressed distance is given by x1*k1 = x2*k2 a) Here we've displaced the block a distance i from its equilibrium, so that x1+x2 = i. Both springs will also be extended (or compressed) in the positive direction. And we'll have: x1*k1 = (i-x1)*k2 x1 =i*k2 / (k1+k2) b) In the same way, we'll have x2 = i*k1 / (k1+k2) c) Force in a spring follows Hooke's law F = -kx For spring 1, we have, unsing answer from a) F1 = -k1 x1 = -i*k1*k2 / (k1+k2) And for F2, we have F2 = -i*k1*k2 / (k1+k2) d) To find the total force, we can use k as defined at the top, so that F = -1 / (1/k1 + 1/k2) * i We also know it'll the sum of F1 and F2 F = -2i*k1*k2 / (k1+k2) A little algebra will convince you that bot expresions are exactly the same. e) The period for an oscillating spring is given by T = 2pv(m/k) = 2pv( m / (1/k1+1/k2) ) f) With springs in parallel, you can simply add the different constants together, so that k = k1 + k2. Here, T = 2pv(m/k) becomes T = 2pv(m/(k1+k2))

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

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