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.
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Step 1: Calculate the angular frequency . The angular frequency for a mass-spring system is given by the formula: = sqrt((k)/(m)) Given m = 2 kg and k = 8 N/m: = sqrt(8 N/m)2 kg = sqrt(4 rad)^2/s^2 = 2 rad/s Step 2: Use the equation of motion for simple harmonic motion (SHM). Assuming the block is released from its maximum displacement A = 2 m at t=0, the position x(t) is given by: x(t) = A ( t) We want to find the time t when x = 1 m. 1 m = (2 m) ((2 rad/s) t) (1)/(2) = ((2 rad/s) t) Step 3: Solve for t. Take the inverse cosine of both sides: ((1)/(2)) = (2 rad/s) t We know that ((1)/(2)) = ()/(3) rad. ()/(3) rad = (2 rad/s) t t = /3 rad2 rad/s = ()/(6) s t ≈ (3.14159)/(6) s ≈ 0.5236 s None of the options (A. 40 sec B. 60 sec C. 30 sec D. 20 sec) match this result. There might be an issue with the question's options or interpretation. However, based on the standard SHM model, this is the time taken. If the question implies traveling to x=1 from x=2 and then back to x=1 or some other interpretation, it's not explicitly stated. Assuming the simplest interpretation of starting at x=2 and moving towards x=0. The final answer is 0.52 s. Step 1: Identify the given frequency. The frequency of the simple pendulum is f = 1.5 Hz. Step 2: Calculate the angular frequency. The relationship between angular frequency and frequency f is: = 2 f Substitute the given frequency: = 2 (1.5 Hz) = 3.0 rad/s The final answer is B. 3.0. Damping of periodic motion always results in a decrease in the amplitude of the oscillations over time due to energy dissipation. The final answer is B. decreasing amplitude. Step 1: Calculate the total mass of the car and passengers. Total mass M = m_car + m_passengers M = 1300 kg + 160 kg = 1460 kg Step 2: Calculate the total effective spring constant. Since there are four springs supporting the car, and they are in parallel, the total effective spring constant k_total is the sum of the individual spring constants. k_total = 4 × k_spring k_total = 4 × 20,000 N/m = 80,000 N/m Step 3: Calculate the angular frequency . The angular frequency for a mass-spring system is given by: = sqrt(k_total)M = sqrt(80,000 N/m)1460 kg = sqrt(54.7945 rad)^2/s^2 ≈ 7.4023 rad/s Step 4: Calculate the frequency f. The frequency f is related to the angular frequency by: f = ()/(2) f = 7.4023 rad/s2 ≈ (7.4023)/(6.28318) Hz ≈ 1.178 Hz Comparing this to the given options: A. 2.31 Hz B. 1.42 Hz C. 5.36 Hz D. 1.18 Hz The calculated frequency is approximately 1.18 Hz. The final answer is D. 1.18 Hz.