An uncharged molecule of DNA (deoxyribonucleic acid) is 2.15 �m long. The ends of the molecule become singly ionized so that there is a charge of −1.6 × 10^−19C on one end
and +1.6 × 10^−19C on the other. The helical molecule acts like a spring and compresses 1.8% upon becoming charged.
Find the effective spring constant of the
molecule. The value of Coulomb’s constant is 8.98755 × 10^9N m2/C2 and the acceleration due to gravity is 9.8m/s2.
Answer in units of N/m
To find the effective spring constant of the DNA molecule, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position.
The formula for Hooke's Law is given as:
F = -kx
Where:
F is the force exerted by the spring,
k is the spring constant, and
x is the displacement from the equilibrium position.
In this case, we are given that the DNA molecule compresses 1.8% upon becoming charged. Since the molecule acts like a spring, we can consider this compression as the displacement.
Let's assume the original length of the molecule before compression is L.
The change in length (ΔL) due to compression is given by:
ΔL = 1.8% * L = 0.018 * L
Now, the displacement x is equal to ΔL:
x = ΔL = 0.018 * L
Substituting this into Hooke's Law, we can express the force in terms of the spring constant:
F = -k * (0.018 * L)
Now, we need to relate the force to the charges on the ends of the molecule and Coulomb's constant.
The electrical force (Fe) between two charges q1 and q2 separated by a distance r is given by Coulomb's Law:
Fe = (k * |q1 * q2|) / r^2
We are given that one end of the molecule has a charge of -1.6 × 10^-19C and the other end has a charge of +1.6 × 10^-19C.
Let's assume the distance between the charges is d.
The force due to the electrical interaction can be expressed as:
F = Fe = (k * |-1.6 × 10^-19 * 1.6 × 10^-19|) / d^2
Since the molecule is compressed, the distance between the charges is reduced. We can relate the original length (L) to the distance between charges (d) using the Pythagorean theorem:
d^2 = L^2 + ΔL^2
Substituting the value of ΔL we previously calculated and rearranging the equation, we have:
d = sqrt(L^2 + (0.018 * L)^2)
Now, equating the forces due to the spring and electrical interaction, we get:
-k * (0.018 * L) = (k * |-1.6 × 10^-19 * 1.6 × 10^-19|) / (sqrt(L^2 + (0.018 * L)^2))^2
Simplifying the equation, we can solve for the spring constant k:
k = ((k * |-1.6 × 10^-19 * 1.6 × 10^-19|) / (sqrt(L^2 + (0.018 * L)^2))^2) / (0.018 * L)
Now, substituting the values of Coulomb's constant (8.98755 × 10^9 N m^2/C^2) and the acceleration due to gravity (9.8 m/s^2), we can calculate the spring constant k in N/m units.
Note: The given value of the acceleration due to gravity is not needed to solve this problem, so it can be disregarded.
Please provide the value of L (original length of the molecule) to proceed with the calculation.