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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You're on a roll — here's the solution for the spring problem: Given: Amplitude, A = 6.0 cm = 0.06 m Period, T = 2.0 s Time, t = 5.2 s Step 1: Calculate the angular frequency (). The angular frequency is related to the period by the formula: = (2)/(T) Substitute the given period: = (2)/(2.0 s) = rad/s Step 2: Compute the position after 5.2 s. Since the mass is released from its maximum displacement (x=A) at t=0, the displacement equation is: x(t) = A ( t) Substitute the values for A and : x(t) = 0.06 ( t) Now, calculate the position at t = 5.2 s: x(5.2) = 0.06 ( × 5.2) x(5.2) = 0.06 (5.2) x(5.2) ≈ 0.06 × (-0.8090) x(5.2) ≈ -0.04854 m The position after 5.2 s is -0.0485 m. Step 3: Compute the velocity after 5.2 s. The velocity is the derivative of the displacement with respect to time: v(t) = (d)/(dt) (A ( t)) = -A ( t) Substitute the values for A and : v(t) = -0.06 ( t) Now, calculate the velocity at t = 5.2 s: v(5.2) = -0.06 ( × 5.2) v(5.2) = -0.06 (5.2) v(5.2) ≈ -0.06 × (-0.5878) v(5.2) ≈ 0.1107 m/s The velocity after 5.2 s is 0.111 m/s. Step 4: Calculate the maximum velocity. The maximum velocity (v_max) for Simple Harmonic Motion is given by: v_max = A Substitute the values for A and : v_max = (0.06 m) × ( rad/s) v_max = 0.06 m