Question

^{Experiments using "optical tweezers" measure the elasticity of individual DNA molecules. For small enough changes in length, the elasticity has the same form as that of a spring. A DNA molecule is anchored at one end, then a force of 1.5 nN (1.5 times 10^-9 N) pulls on the other end, causing the molecule to stretch by 5.0 nm (5.0 times 10^-9 m),as shown in the figure.(p.s. The source of the external force is the laser beam that traps and moves the polystyrene bead in the horizontal direction.) (a) Find the spring constant of this DNA molecule. What is the direction of the restoring force when the DNA molecule is stretched? (b) Find the magnitude and direction of the restoring force while this DNA molecule is suppressed by 7.0 nm. Equation Sheet W = mg; f_s lessthanorequalto mu_s N = f_smax; f_k = mu_k N v = r omega; omega = delta theta/delta t; alpha = delta omega/delta t; a_t = r alpha; a_c = v^2/r; tau = Plusminus r F sin phi; I =Sigma_i m_i r^2_i; tau = I alpha Equilibrium: Sigma F_X = 0; Sigma F_y = 0; tau_net = 0; F_spx = -k delta x Universal constant: g = 9.8 m/s^2 A vector = (A_x, A_y) rightarrow A_x = |A vector|cos theta;A_y = |A vector| sin theta; |A vector|= squareroot A^2_x + A^2_y;theta = tan^-1 (A_y/A_x) A: any physical quantity such as position, velocity (v), acceleration (alpha), force (F) ellipsis etc. P vector = mv vector;= delta p vector= p_f vector - p_i vector = m delta v vector = m( v_f vector - v_i vector) = F_avg vector delta t (or 0) For all collisions : p_i total vector = p_f total vector = p_i vector + p_2 vector + ellipsis = constant}

Answer 1

(a) the spring constant of the DNA molecule is given by F = kx k = F/x = 1.5 times 10^-9 N/5.0 times 10^-9 m = 0.3 N/m the direction of the restoring force towards left side. (b) the magnitude of restoring force is, F = kx = (0.3 N/m)(7.0 times 10^-9 m) = 2.1 times 10^-9 N the direction of the force along right.