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.

The stage moves at a constant speed while stretching the DNA. Which of the graphs in Fig. given below best represents the power supplied to the stage versus time?

Step by step solution

01

Relationship between power and time

We know that the power can be written as product of the force and velocity

\( \Rightarrow P = F \cdot v\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)\)

Also, by Hooke’s law, we know that

\(F = kx\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 2 \right)\)

So, by equations (1) and (2)

\( \Rightarrow P = kx \cdot v\)

02

Identification of given data

Here we have given that, the stage moves at a constant speed while stretching the DNA.

03

Finding the graph of power versus time

Here by step 1 we have\(P = kxv\)

Also, we have speed is constant

So, power and displacement are in direct proportionality.

So we can say that the power increases linearly when time increases.

So, the graph given in option (a) is correct.

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