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Consider two massless springs connected in series. Spring 1 has a spring constant k1, and spring 2 has a spring constant k2. A constant force of magnitude F is being applied to the right. When the two springs are connected in this way, they form a system equivalent to a single spring of spring constant k.

Part A

What is the effective spring constant k of the two-spring system? Express the effective spring constant in terms of k1 and k2.

Part B

Now consider three springs set up in series as shown. (Figure 2) The spring constants are k1, k2, and k3, and the force acting to the right again has magnitude F. Find the spring constant k′ of the three-spring system. Express your answer in terms of k1, k2, and k3.

Respuesta :

oladayoademosu

Answer:

a. k = (1/k₁ + 1/k₂)⁻¹ b. k = (1/k₁ + 1/k₂ + 1/k₃)⁻¹

Explanation:

Since only one force F acts, the force on spring with spring constant k₁ is F = k₁x₁ where x₁ is its extension

the force on spring with spring constant k₂ is F = k₂x₂ where x₁ is its extension

Let F = kx be the force on the equivalent spring with spring constant k and extension x.

The total extension , x = x₁ + x₂

x = F/k = F/k₁ + F/k₂

1/k = 1/k₁ + 1/k₂

k = (1/k₁ + 1/k₂)⁻¹

B

The force on spring with spring constant k₃ is F = k₃x₃ where x₃ is its extension

Let F = kx be the force on the equivalent spring with spring constant k and extension x.

The total extension , x = x₁ + x₂ + x₃

x = F/k = F/k₁ + F/k₂ + F/k₃

1/k = 1/k₁ + 1/k₂ + 1/k₃

k = (1/k₁ + 1/k₂ + 1/k₃)⁻¹

annyksl

The effective spring constant k of the two-spring system will be equal to (1/k₁ + 1/k₂)⁻¹, while the spring constant k′ of the three-spring system will be equal to (1/k₁ + 1/k₂ + 1/k₃)⁻¹.

We can arrive at this answer as follows:

  • We assume that the constants related to the force of spring [tex]K_1[/tex] is equal to [tex]K_1*X_1[/tex], while the constants of spring [tex]K_2[/tex] are equal to [tex]K_2*X_2[/tex].

We must consider the variable X as the extension of each spring.

  • In this case, we must consider that the formula for the full extension will be:

[tex]x = x_1 + x_2[/tex]

  • Solving this equation we will have:

[tex]x=\frac{F}{K}=\frac{F}{K_1} +\frac{F}{K_2} \\\frac{1}{K} =\frac{1}{K_1} +\frac{1}{K_2}\\K= (\frac{1}{K_1} +\frac{1}{K_2})^-1[/tex]

In this way, we were able to find the effective spring constant k of the two-spring system.

Continuing, we should consider the spring constants K3 being represented by [tex]K_3*X_3[/tex], where the X also represents the spring length.

  • Therefore, we can consider the total extension equal to:

[tex]x=x_1+x_2+x_3\\x= \frac{F}{K} = \frac{F}{K_2} +\frac{F}{K_2} +\frac{F}{K_3} \\\frac{1}{K} =\frac{1}{K_2} +\frac{1}{K_2} +\frac{1}{K_3}\\K=(\frac{1}{K_2} +\frac{1}{K_2} +\frac{1}{K_3})^-1[/tex]

In this way, we were able to find the spring constant k′ of the three-spring system.

More information about what is a constant:

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