Respuesta :

The spring with twice spring constant has to compress [tex]\boxed{\frac{1}{{\sqrt 2 }}}[/tex] or [tex]\boxed{0.707}[/tex] times the compression of spring of the given spring constant.

Further explanation:

Consider a spring of spring constant k is compressed by length [tex]x[/tex].

Then amount of energy stored in the spring 1 can be calculated as.

[tex]{E_1}=\dfrac{1}{2}k{x^2}[/tex]                                                                 …… (1)

Here, [tex]{E_1}[/tex] is the amount of energy store in spring 1 due to its compression.

Now, another spring with spring constant [tex]2k[/tex] as given is compressed by length [tex]y[/tex] .

Then the energy stored in the spring due to its compression can be calculated as.

[tex]{E_2}=\dfrac{1}{2}\left( {2k}\right){y^2}[/tex]                            …… (2)

Here, [tex]{E_2}[/tex] is the amount of energy store in spring 2 due to its compression.

As given in the question energy store in both the spring is same. So, equate equation (1) and equation (2).

[tex]\begin{aligned}\frac{1}{2}k{x^2}&=\frac{1}{2}\left( {2k} \right){y^2}\\k{x^2}&=\left( {2k}\right){y^2}\\\frac{1}{2}{x^2}&={y^2}\\\end{aligned}[/tex]

   

Taking square root both the sides in above equation,

[tex]\begin{gathered}\sqrt {\frac{1}{2}{x^2}}=\sqrt {{y^2}}\\\frac{x}{{\sqrt 2 }} = y \\ \end{gathered}[/tex]

 

The above equation can be written as,

[tex]\boxed{y=\frac{x}{{\sqrt 2 }}}[/tex]

 

So, spring of spring constant [tex]2k[/tex] is compressed [tex]\dfrac{1}{{\sqrt 2 }}[/tex] times the compression of spring of spring constant [tex]k[/tex].

Learn more:

1. Conservation of energy: brainly.com/question/3943029.

2. Motion under friction https://brainly.com/question/7031524.

3. Conservation of momentum https://brainly.com/question/9484203.

Answer detail:

Grade: Senior school

Chapter: Work and Energy

Subject: Physics

Keywords:

Compression, expansion, spring, twice the spring constant, same amount of energy, spring constant, energy, potential energy, kinetic energy, force, Hook’s law.

The distance by which you compress a spring with twice the spring constant to store the same amount of energy is;

0.7071 times the initial distance of compression.

Formula for Energy in a spring is;

E = ¹/₂kx²

where;

x is distance by which spring is compressed

k is spring constant

Now, for the original spring which we will call spring 1, we will have;

E₁ = ¹/₂kx²

Now, we are told that the spring constant is now doubled for a new spring which we will call spring 2. Thus;

E₂ =  ¹/₂(2k)y²

where y is distance by which spring is compressed.

Now for spring 2 to store the same amount of energy as spring 1, then;

E₁ = E₂

Thus;

¹/₂kx² = ¹/₂(2k)y²

Common terms will cancel out to give;

x² = 2y²

y² = x²/2

y = x/√2

y = 0.7071x

    In conclusion, the new spring must be compressed by a multiplicative factor of 0.7071 of the given spring distance of compression.

Read more at; https://brainly.com/question/22525611