A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has v_x = -30 cm/s. Determine: The maximum speed.

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Convert the given values into SI units: mass (m) = 200 g = 0.2 kg, frequency (f) = 2.0 Hz, initial position (x₀) = 5.0 cm = 0.05 m, and initial velocity (vₓ₀) = -30 cm/s = -0.3 m/s.

Recall the relationship between angular frequency (ω) and frequency (f): ω = 2πf. Substitute f = 2.0 Hz into the formula to calculate ω.

The maximum speed (vₘₐₓ) in simple harmonic motion is given by the formula vₘₐₓ = Aω, where A is the amplitude of oscillation. To find A, use the equation for total energy in simple harmonic motion: E = (1/2)mω²A² = (1/2)m(vₓ₀² + ω²x₀²). Solve for A using the given initial conditions.

Once A is determined, substitute its value and the calculated ω into the formula vₘₐₓ = Aω to find the maximum speed.

Ensure all units are consistent and verify the calculations conceptually by checking that the maximum speed occurs when the mass passes through the equilibrium position (x = 0).

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주요 개념

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Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In SHM, the restoring force is directly proportional to the displacement from the equilibrium and acts in the opposite direction. The motion can be described by sinusoidal functions, and key parameters include amplitude, frequency, and maximum speed.

Maximum Speed in SHM

The maximum speed of an object in Simple Harmonic Motion occurs as it passes through the equilibrium position. It can be calculated using the formula v_max = ωA, where ω is the angular frequency (ω = 2πf) and A is the amplitude of the motion. This speed is crucial for understanding the dynamics of oscillating systems.

Frequency and Angular Frequency

Frequency is the number of oscillations per unit time, measured in Hertz (Hz). Angular frequency, denoted as ω, relates to frequency by the equation ω = 2πf. Understanding the relationship between frequency and angular frequency is essential for calculating various properties of oscillating systems, including maximum speed and energy.