Question

Consider two massless springs connected in parallel. Springs 1 and 2 have spring constants k1 and k2 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 k1 and k2.Part B:Now consider three springs connected in parallel. The spring constants of springs 1, 2, and 3 are k1, k2, and k3. The springs are connected by a vertical rod, and a force of magnitude F is being exerted to the right.Find the effective spring constant k? of the three-spring system.Give your answer in terms of k1, k2, and k3.

Answer 1

Answer:

Part A : k = k₁+k₂

Part B: k = k₁+k₂+.....kₙk₃

Where for 3 springs n= 3

Explanation:

Hooke's law states that the force required to stretch a spring is directly proportional to the distance x, where k is the spring constant.

Mathematically, Hooke's law is represented as

F = kx

Where F = Force required to stretch the spring

k = spring constant

x = distance

Part A:

Since the two massless springs in Part A are connected in parallel, it means they are stretched by the same Force (F)

Therefore using Hooke's law

F = kx ......... Equation 1

Total Force is represented by F

Spring 1 , F₁= k₁x ....... Equation 2

Spring 2, F₂= k₂x ......... Equation 3

Total Force = F₁ + F₂ .......... Equation 4

We substitute k₁x for F₁ and k₂x for F₂ in equation 4

F= k₁x+k₂x ........ Equation 5

F = (k₁+k₂)x ........ Equation 6

According to Hooke's law, F = kx

We substitute kx for F in Equation 6

Hence, kx = (k₁+k₂)x ........... Equation 7

Therefore, the Effective springs constant for two massless spring connected in parallel is given as

k = k₁+k₂