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Chapter 7: Problem 18

A biophysicist grabs the ends of a DNA strand with optical tweezers and stretches it \(26 \mu \mathrm{m}\). How much energy is stored in the stretched molecule if its spring constant is \(0.046 \mathrm{pN} / \mu \mathrm{m} ?\)

Short Answer

Expert verified
The energy stored in the DNA strand when stretched is found by substituting the values of the spring constant and displacement into the formula for elastic potential energy. Upon calculation, the energy comes out to be approximately \( 1.548 \times 10^{-20} \) Joules.

Step by step solution

01

Convert Units

The spring constant given is in \( \mathrm{pN}/\mu \mathrm{m} \) and displacement \( x \) is in \( \mu \mathrm{m} \). But the SI unit of force is Newton (N) and that of displacement is meter (m). So we need to convert \( \mathrm{pN} \) to \( \mathrm{N} \) and \( \mu \mathrm{m} \) to \( \mathrm{m} \) before substituting the values into the formula. \1 piconewton (pN) = \(1 \times 10^{-12} \) Newtons (N) \Therefore, \( k = 0.046 \times 10^{-12} \) N/m \1 micrometer (\( \mu \mathrm{m} \)) = \( 1 \times 10^{-6} \) meters (m) \Therefore, \( x = 26 \times 10^{-6} \) m
02

Calculate energy

Substitute \( k \) and \( x \) into the formula \( U = \frac{1}{2} k x^2 \). \The elastic potential energy stored in the DNA strand will be \\( U = 0.5 \times (0.046 \times 10^{-12} \) N/m\() \times (26 \times 10^{-6} \) m\()^2 \)

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Most popular questions from this chapter

Find the potential energy of a 70 -kg hiker (a) atop New Hampshire's Mount Washington, \(1900 \mathrm{m}\) above sea level, and (b) in Death Valley, California, 86 m below sea level. Take the zero of potential energy at sea level.

A skier starts down a frictionless \(32^{\circ}\) slope. After a vertical drop of \(25 \mathrm{m},\) the slope temporarily levels out and then slopes down at \(20^{\circ},\) dropping an additional 38 m vertically before leveling out again. Find the skier's speed on the two level stretches.

A spring of constant \(k=340 \mathrm{N} / \mathrm{m}\) is used to launch a \(1.5-\mathrm{kg}\) block along a horizontal surface whose coefficient of sliding friction is \(0.27 .\) If the spring is compressed \(18 \mathrm{cm},\) how far does the block slide?

A particle of mass \(m\) is subject to a force \(\vec{F}=(a \sqrt{x}) \hat{\imath},\) where \(a\) is a constant. The particle is initially at rest at the origin and is given a slight nudge in the positive \(x\) -direction. Find an expression for its speed as a function of position \(x\)

As an energy-efficiency consultant, you're asked to assess a pumped-storage facility. Its reservoir sits \(140 \mathrm{m}\) above its generating station and holds \(8.5 \times 10^{9} \mathrm{kg}\) of water. The power plant generates 330 MW of electric power while draining the reservoir over an 8.0 -h period. Its efficiency is the percentage of the stored potential energy that gets converted to electricity. What efficiency do you report?
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