This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Establish the relationship between frequency, tension, length, and diameter. Let f be the frequency, T be the tension, L be the length, and d be the diameter. The problem states that the frequency f is directly proportional to the square root of the tension T, and inversely proportional to the length L and the diameter d. This can be written as: f = k sqrt(T)Ld where k is the constant of proportionality. Step 2: Express the changes in T, L, and d. Let the initial values be f_1, T_1, L_1, d_1. The initial frequency is: f_1 = k sqrt(T_1)L_1 d_1 The new values are: • Tension T is increased by 30%: T_2 = T_1 + 0.30 T_1 = 1.30 T_1 • Length L is halved: L_2 = (1)/(2) L_1 = 0.5 L_1 • Diameter d is halved: d_2 = (1)/(2) d_1 = 0.5 d_1 Step 3: Calculate the new frequency f_2. Substitute the new values into the relationship: f_2 = k sqrt(T_2)L_2 d_2 f_2 = k sqrt(1.30 T_1)(0.5 L_1)(0.5 d_1) f_2 = k sqrt(1.30) sqrt(T_1)0.25 L_1 d_1 Rearrange the terms to relate f_2 to f_1: f_2 = (sqrt(1.30)0.25) (k sqrt(T_1)L_1 d_1) Since f_1 = k sqrt(T_1)L_1 d_1, we have: f_2 = sqrt(1.30)0.25 f_1 Calculate the numerical factor: sqrt(1.30)0.25 ≈ (1.140175)/(0.25) ≈ 4.5607 So, f_2 ≈ 4.5607 f_1. Step 4: Calculate the percentage change in frequency. The percentage change is given by: Percentage Change = (f_2 - f_1)/(f_1) × 100\% Substitute f_2 ≈ 4.5607 f_1: Percentage Change = (4.5607 f_1 - f_1)/(f_1) × 100\% Percentage Change = ((4.5607 - 1) f_1)/(f_1) × 100\% Percentage Change = 3.5607 × 100\% Percentage Change = 356.07\% Since the value is positive, it is a percentage increase. The percentage change in the frequency is approximately 356.1% increase. 356.1\% increase 4 down, 1 to go! What's next?