Question

How far must you compress a spring with twice the spring constant to store the same amount of energy?

Answer 1

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

Further explanation:

Consider a spring of spring constant k is compressed by length $x$.

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

${E_1}=\dfrac{1}{2}k{x^2}$                                                                 …… (1)

Here, ${E_1}$ is the amount of energy store in spring 1 due to its compression.

Now, another spring with spring constant $2k$ as given is compressed by length $y$ .

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

${E_2}=\dfrac{1}{2}\left( {2k}\right){y^2}$                            …… (2)

Here, ${E_2}$ 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).

$\begin{aligned}\frac{1}{2}k{x^2}&=\frac{1}{2}\left( {2k} \right){y^2}\\k{x^2}&=\left( {2k}\right){y^2}\\\frac{1}{2}{x^2}&={y^2}\\\end{aligned}$

   

Taking square root both the sides in above equation,

$\begin{gathered}\sqrt {\frac{1}{2}{x^2}}=\sqrt {{y^2}}\\\frac{x}{{\sqrt 2 }} = y \\ \end{gathered}$

 

The above equation can be written as,

$\boxed{y=\frac{x}{{\sqrt 2 }}}$

 

So, spring of spring constant $2k$ is compressed $\dfrac{1}{{\sqrt 2 }}$ times the compression of spring of spring constant $k$.

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.

Answer 2

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.

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