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.

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

A) The effective spring constant in terms of k₁ and k₂ is; k = 1/[(1/k₁) + (1/k₂)]

B) The effective spring constant in terms of k₁, k₂ and k₃ is; k = 1/[(1/k₁) + (1/k₂) + (1/k₃)]

A) We are told that springs 1 and 2 have spring constant as k₁ and k₂ respectively.

Now, formula for force on a spring is;

F = kx (the negative sign was ignored because it only shows that the force of restoration acts in an opposite direction to the applied force)

Where;

F is force

k is spring constant

x is extension of spring.

Thus;

F1 = k₁•x₁

F2 = k₂•x₂

Now, we are told that the springs are extended by the same amount.

Thus, total extension;

x_eq = x₁ + x₂

Since a constant force is being exerted on the rod, then from F = kx, we have;

F/k = F₁/k₁ + F₂/k₂

However, F₁ = F₂ = F since we are told that the same force acts on the rod. Thus;

F/k = F/k₁ + F/k₂

F will cancel out to give;

1/k = (1/k₁) + (1/k₂)

k = 1/[(1/k₁) + (1/k₂)]

B) We are told that there is now a third spring connected in parallel with the other 2 springs.

Just like in answer A above, force on the third spring will be;

F₃ = k₃x₃

Since the force is constant throughout, it means it's the same as for the other 2 springs in the system. Thus;

F/k = F/k₁ + F/k₂ + F/k₃

F will cancel out to give;

1/k = (1/k₁) + (1/k₂) + (1/k₃)

We want to make k this subject of the formula. Thus;

k = 1/[(1/k₁) + (1/k₂) + (1/k₃)]

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