Question

C. Springs A and B have spring constants k, and k₂ respectively. It is known that k₂ is twice k₁. The springs are connected in parallel between two plates.

i. Calculate the effective spring constant of the system in terms of k₁.
ii. If the springs store energy E₁ and E2 respectively, determine the value of the ration E1/E2​

Answer 1

i. The effective spring constant of the system can be calculated by adding the reciprocals of the spring constants.

ii. The ratio of the energy stored in spring A to spring B is 1/8.

Step-by-step explanation:

i. The effective spring constant of the system in terms of k₁ can be calculated by adding the reciprocals of the spring constants.

So, the effective spring constant (k_eff) is given by: k_eff = 1/(1/k₁ + 1/k₂).

Since k₂ is twice k₁, we can substitute it into the equation: k_eff = 1/(1/k₁ + 1/2k₁).

Simplifying this expression gives us: k_eff = 2k₁/3.

ii. The ratio of the energy stored in spring A (E₁) to the energy stored in spring B (E₂) is given by: E₁/E₂ = (1/2)k₁x₁² / (1/2)(2k₁)x₂² = (1/2)k₁x₁² / k₁x₂² = x₁² / 2x₂².

Since k₂ is twice k₁, we can substitute it into the equation: E₁/E₂ = x₁² / 2(2x₁)² = x₁² / 8x₁² = 1/8.