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A DNA molecule, with its double- helix structure, can in some situations behave like a spring. Measuring the force required to stretch single DNA molecules under various conditions can provide information about the biophysical properties of DNA. A technique for measuring the stretching force makes use of a very small cantilever, which consists of a beam that is supported at one end and is free to move at the other end, like a tiny diving board. The cantilever is constructed so that it obeys Hooke’s law—that is, the displacement of its free end is proportional to the force applied to it. Because different cantilevers have different force constants, the cantilever’s response must first be calibrated by applying a known force and determining the resulting deflection of the cantilever. Then one end of a DNA molecule is attached to the free end of the cantilever, and the other end of the DNA molecule is attached to a small stage that can be moved away from the cantilever, stretching the DNA. The stretched DNA pulls on the cantilever, deflecting the end of the cantilever very slightly. The measured deflection is then used to determine the force on the DNA molecule.

A segment of DNA is put in place and stretched. Given figure shows a graph of the force exerted on the DNA as a function of the displacement of the stage. Based on this graph, which statement is the best interpretation of the DNA’s behaviour over this range of displacements? The DNA

1. Does not follow Hooke’s law, because its force constant increases as the force on it increases
2. Follows Hooke’s law and has a force constant of about\(0.1{\rm{ pN/nm}}\)
3. Follows Hooke’s law and has a force constant of about\(10{\rm{ pN/nm}}\)
4. Does not follow Hooke’s law, because its force constant decreases as the force on it increase

Step by step solution

01

Hooke’s law

With the aid of the subsequent equation, Hooke's law calculates the force applied by the spring to the object attached to it.

\(F = - kx\).

Where k is the spring constant.

02

Identification of given data

Here we have graph of force versus displacement. Which is given by

From that we can find value of force and displacement.

03

Finding the suitable option for the given graph.

Here from the graph, we can see that force is directly proportional to displacement.

So, if distance increasing, force is increasing.

So, we can say that it follow’s the Hooke’s law.

Also, we can find the spring constant.

\(\begin{aligned}{} \Rightarrow F{\rm{ = kx}}\\ \Rightarrow {\rm{k = }}\frac{F}{x}\end{aligned}\)

\(\begin{aligned}{} \Rightarrow k = \frac{{{\rm{5 pN}}}}{{{\rm{50 nm}}}}\\ \Rightarrow k = 0.1{\rm{ pN/nm}}\end{aligned}\)

Hence , It Follows Hooke’s law and has a force constant of about \(0.1{\rm{ pN/nm}}\)

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