(i) The spring constant (k) can be determined using Hooke's Law, which states that the force exerted by a spring is equal to the displacement from its equilibrium position times the spring constant. In mathematical terms, F = kx. Here, the force (F) is the weight of the mass, which is its mass times gravity (1.00 kg * 9.8 m/s^2 = 9.8 N), and the displacement (x) is the amount the spring is compressed, which is 12 mm or 0.012 m. Solving for k gives k = F/x = 9.8 N / 0.012 m = 816.67 N/m.

(ii) The elastic potential energy (U) stored in a spring is given by the formula U = 1/2 kx^2. Substituting the values we have, U = 1/2 * 816.67 N/m * (0.012 m)^2 = 0.059 Nm or 0.059 J.

(iii) The energy stored in the spring came from the work done to compress the spring. This work was done by the force of gravity acting on the mass as it was placed on the spring.

(iv) The frequency of oscillation (f) of a mass-spring system is given by the formula f = 1/(2π) * sqrt(k/m), where k is the spring constant and m is the mass. Substituting the values we have, f = 1/(2π) * sqrt(816.67 N/m / 0.74 kg) = 1.39 Hz.

(v) The period of oscillation (T) of a pendulum is given by the formula T = 2π * sqrt(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Since the pendulum is oscillating in time with the mass-spring system, its period is the same as the period of the mass-spring system, which is the reciprocal of the frequency, or T = 1/f = 1/1.39 Hz = 0.72 s. Solving the pendulum period formula for L gives L = (T^2 * g) / (4π^2) = (0.72 s)^2 * 9.8 m/s^2 / (4π^2) = 0.53 m.

Question

(i) The spring constant (k) can be determined using Hooke's Law, which states that the force exerted by a spring is equal to the displacement from its equilibrium position times the spring constant. In mathematical terms, F = kx. Here, the force (F) is the weight of the mass, which is its mass times gravity (1.00 kg * 9.8 m/s^2 = 9.8 N), and the displacement (x) is the amount the spring is compressed, which is 12 mm or 0.012 m. Solving for k gives k = F/x = 9.8 N / 0.012 m = 816.67 N/m.

(ii) The elastic potential energy (U) stored in a spring is given by the formula U = 1/2 kx^2. Substituting the values we have, U = 1/2 * 816.67 N/m * (0.012 m)^2 = 0.059 Nm or 0.059 J.

(iii) The energy stored in the spring came from the work done to compress the spring. This work was done by the force of gravity acting on the mass as it was placed on the spring.

(iv) The frequency of oscillation (f) of a mass-spring system is given by the formula f = 1/(2π) * sqrt(k/m), where k is the spring constant and m is the mass. Substituting the values we have, f = 1/(2π) * sqrt(816.67 N/m / 0.74 kg) = 1.39 Hz.

(v) The period of oscillation (T) of a pendulum is given by the formula T = 2π * sqrt(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Since the pendulum is oscillating in time with the mass-spring system, its period is the same as the period of the mass-spring system, which is the reciprocal of the frequency, or T = 1/f = 1/1.39 Hz = 0.72 s. Solving the pendulum period formula for L gives L = (T^2 * g) / (4π^2) = (0.72 s)^2 * 9.8 m/s^2 / (4π^2) = 0.53 m.

Knowee AI · Accepted Answer

(i) The spring constant (k) can be determined using Hooke's Law, which states that the force exerted by a spring is equal to the displacement from its equilibrium position times the spring constant. In mathematical terms, F = kx. Here, the force (F) is the weight of the mass, which is its mass times gravity (1.00 kg * 9.8 m/s^2 = 9.8 N), and the displacement (x) is the amount the spring is compressed, which is 12 mm or 0.012 m. Solving for k gives k = F/x = 9.8 N / 0.012 m = 816.67 N/m.

(ii) The elastic potential energy (U) stored in a spring is given by the formula U = 1/2 kx^2. Substituting the values we have, U = 1/2 * 816.67 N/m * (0.012 m)^2 = 0.059 Nm or 0.059 J.

(iii) The energy stored in the spring came from the work done to compress the spring. This work was done by the force of gravity acting on the mass as it was placed on the spring.

(iv) The frequency of oscillation (f) of a mass-spring system is given by the formula f = 1/(2π) * sqrt(k/m), where k is the spring constant and m is the mass. Substituting the values we have, f = 1/(2π) * sqrt(816.67 N/m / 0.74 kg) = 1.39 Hz.

(v) The period of oscillation (T) of a pendulum is given by the formula T = 2π * sqrt(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Since the pendulum is oscillating in time with the mass-spring system, its period is the same as the period of the mass-spring system, which is the reciprocal of the frequency, or T = 1/f = 1/1.39 Hz = 0.72 s. Solving the pendulum period formula for L gives L = (T^2 * g) / (4π^2) = (0.72 s)^2 * 9.8 m/s^2 / (4π^2) = 0.53 m.

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