Question

Be the action of a force of 51N, a spring measures 39cm. Its length becomes 40.8 cm when subjected to another force of 61N. 1)Determine the empty length of the spring 2)Determine an elongation which will correspond to a force of 32N.3) So what is its length

Answer 1

1) 29.82 cm

2) 5.76 cm

3) 35.58 cm

Step-by-step explanation:

Part 1)

The force of a spring is equal to:

F = kΔx

Where k is the constant of the spring and Δx is the elongation. Δx = xf - xi, where xf is the length of the spring when the force is applied and xi is the empty length. Then

F = k(xf - xi)

Now, by the action of a force of 51N, a spring measures 39 cm, so

51 = k(39 - xi)

And by the action of a force of 61N, the spring length is 40.8 cm, so

61 = k(40.8 - xi)

To find the empty length, we need to solve the system of equations

51 = k(39 - xi)

61 = k(40.8 - xi)

First, solve the first equation for k

`k=(51)/(39-x_i)`

Then, replace this on the second equation and solve for xi

`\begin{gathered} 61=k(40.8-x_i) \\ 61=(51)/((39-x_i))(40.8-x_i) \\ 61(39-x_i)=51(40.8-x_i) \\ 61(39)-61(x_i)=51(40.8)-51(x_i) \\ 2379-61x_i=2080.8-51x_i \\ 2379-2080.8=61x_i-51x_i \\ 298.2=10x_i \\ (298.2)/(10)=x_i \\ 29.82=x_i \end{gathered}`

Therefore, the empty length of the spring is 29.82 cm

Part 2)

Now, we need to calculate the value of k, so replacing xi = 29.82, we get:

`k=(51)/(39-29.82)=5.556`

Therefore, the equation for the force is

F = 5.556Δx

Solving for Δx, we get:

Δx = F/5.556

Replacing the force by 32N, we can calculate the elongation as

Δx = 32/5.556 = 5.76 cm

Part 3)

Then, the length can be calculated by solving the following equation for xf

Δx = xf - xi

xf = Δx + xi

Replacing Δx = 5.76 cm and xi = 29.82 cm, we get:

xf = 5.76 cm + 29.82 cm

xf = 35.58 cm

So, its length is 35.58 cm

Therefore, the answers are

1) 29.82 cm

2) 5.76 cm

3) 35.58 cm