Verified Answer :
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 \(k\).
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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.
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