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 position at t = 0.40 s.

검증된 단계별 안내

Convert all given quantities to 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 = -0.30 m/s.

Determine the angular frequency (ω) of the oscillation using the formula:

ω = 2 π f

. Substitute f = 2.0 Hz to calculate ω.

Write the general equation for the position of a mass in simple harmonic motion:

x (t) = A cos(ωt + ϕ)

, where A is the amplitude and ϕ is the phase constant. Use the initial conditions to determine A and ϕ.

Use the initial position (x₀ = 0.05 m) and initial velocity (vₓ₀ = -0.30 m/s) to solve for A and ϕ. Start by substituting x₀ into the position equation and vₓ₀ into the velocity equation:

v (t) = - A ω sin(ωt + ϕ)

. Solve the system of equations to find A and ϕ.

Substitute the values of A, ω, and ϕ into the position equation

x (t) = A cos(ωt + ϕ)

. Evaluate the position at t = 0.40 s to find x(0.40 s).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion is characterized by a restoring force proportional to the displacement from the equilibrium, leading to sinusoidal position and velocity functions over time. In this case, the mass-spring system exhibits SHM, which can be described using parameters like amplitude, frequency, and phase.

Frequency and Angular Frequency

Frequency is the number of oscillations per unit time, measured in Hertz (Hz). Angular frequency, denoted by ω, relates to frequency through the equation ω = 2πf, where f is the frequency. In this problem, the frequency of 2.0 Hz indicates how quickly the mass oscillates, which is crucial for determining its position at any given time.

Position and Velocity in SHM

In SHM, the position (x) and velocity (v) of the oscillating mass can be expressed as functions of time. The position can be described by the equation x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. The velocity is the derivative of the position function, indicating how the position changes over time, which is essential for calculating the position at a specific time.