A 350 g mass on a 45-cm-long string is released at an angle of 4.5° from vertical. It has a damping constant of 0.010 kg/s. After 25 s, (a) how many oscillations has it completed and (b) what fraction of the initial energy has been lost?
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Step 1: Calculate the natural frequency of the pendulum using the formula , where is the acceleration due to gravity (9.8 m/s²) and is the length of the string (0.45 m).
Step 2: Determine the damping ratio using the formula , where is the damping constant (0.010 kg/s) and is the mass (0.350 kg).
Step 3: Calculate the damped angular frequency using the formula .
Step 4: Find the number of oscillations completed in 25 seconds using the formula , where is the time (25 s).
Step 5: Calculate the fraction of the initial energy lost using the formula , where is the initial energy and is the energy after time . The fraction lost is .