The angular displacement of a simple pendulum is given by θ(t)= θ_m cos (ωt + φ).If the pendulum is 45 cm in length, and is given an angular speed dθ/dt = 3.4 rad/s at time t = 0, when it is hanging vertically, what is θ_m? A)4.6 rad B)3.4 rad C)1.4 rad D)0.73 rad E)0.45 rad

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Quizplus · Accepted Answer

Given the angular speed (dθ/dt) at t = 0 is 3.4 rad/s and the pendulum is hanging vertically (θ = 0), we can use the formula for angular speed: dθ/dt = -ωθ_msin(ωt + φ). At t = 0, φ = 0 (since it's hanging vertically and we're considering this initial position), so dθ/dt = -ωθ_msin(φ). Given dθ/dt = 3.4 rad/s and sin(φ) = sin(0) = 0, we need to find θ_m using the length of the pendulum to find ω. The formula for ω in a pendulum is ω = sqrt(g/L), where g is the acceleration due to gravity (9.8 m/s^2) and L is the length of the pendulum in meters (0.45 m). Calculating ω gives ω = sqrt(9.8/0.45). However, since the initial condition given does not directly involve ω for calculating θ_m and the angular speed is given at the moment it's vertical, we need to reconsider the approach using the given information directly related to θ_m and the initial angular speed. The initial condition implies a relationship between the maximum angular displacement and the angular speed, but the direct calculation from the given information was misunderstood in the explanation. The correct approach involves understanding that at t = 0, the cosine term in the displacement equation would be cos(φ), and since the pendulum is at its initial position hanging vertically, we look at the conditions for maximum angular displacement and its relation to angular speed without directly involving the pendulum length or gravity in the calculation. The mistake was in the direct application of formulas without correctly interpreting the given conditions for θ_m. The correct interpretation should focus on the given angular speed and how it relates to the maximum angular displacement (θ_m) through the initial conditions provided.

The Angular Displacement of a Simple Pendulum Is Given by θ(T)=