Physics, 28.01.2020 13:49 cxttiemsp021

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

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Answer from: misk980

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 .

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 as given is compressed by length .

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

Learn more:

1. Conservation of energy: link.

2. Motion under friction link.

3. Conservation of momentum link.

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 from: Quest

answer would be letter choice (a), a ball resting on a shelf. would be another example of potential energy > stored energy > not moving.

a ball spinning on a finger > kinetic energy > moving energy

a ball rolling on the ground > kinetic energy > moving energy

a ball flying through the air > kinetic energy > moving energy

hope that : )

Answer from: Quest

the chart show the masses and velocities of two colliding objects that stick together after a collision.

according to the law of conservation of momentum, the momentum of the object after the collision is

answer

1,500

kg • m/s.

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