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 vₓ = -30 cm/s. Determine: The total energy.

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Convert the given values into SI units: mass (m) = 200 g = 0.200 kg, displacement (x) = 5.0 cm = 0.050 m, velocity (vₓ) = -30 cm/s = -0.30 m/s, and frequency (f) = 2.0 Hz.

Calculate the angular frequency (ω) using the formula: ω = 2πf. Substituting f = 2.0 Hz, we get ω = 2π × 2.0 rad/s.

The total energy (E) in a simple harmonic oscillator is given by the sum of the kinetic energy (K) and potential energy (U): E = K + U. The kinetic energy is K = (1/2)mvₓ², and the potential energy is U = (1/2)kx², where k is the spring constant.

Determine the spring constant (k) using the relationship between angular frequency and the spring constant: ω = √(k/m). Rearrange to find k: k = mω². Substitute the values of m and ω to calculate k.

Substitute the values of m, vₓ, k, and x into the expressions for K and U, then add them to find the total energy: E = (1/2)mvₓ² + (1/2)kx².

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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 phase. Understanding SHM is essential for analyzing systems like springs and pendulums.

Mechanical Energy in Oscillatory Systems

In oscillatory systems, mechanical energy is conserved and can be expressed as the sum of kinetic and potential energy. For a mass-spring system, the total mechanical energy (E) is given by E = 1/2 k A², where k is the spring constant and A is the amplitude. At any point in the oscillation, the energy can be divided into kinetic energy (due to motion) and potential energy (due to displacement from equilibrium). This concept is crucial for determining the total energy of the system.

Frequency and Angular Frequency

Frequency is the number of oscillations per unit time, measured in Hertz (Hz), and is related to the period of the motion. Angular frequency (ω) is a measure of how quickly the oscillation occurs, defined as ω = 2πf, where f is the frequency. In the context of oscillatory motion, knowing the frequency allows us to calculate other properties of the motion, such as the maximum velocity and energy, which are essential for solving problems related to oscillating systems.