The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t - π/4), where t is in s. Determine: The initial conditions.

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Step 1: Understand the given equation for the position of the oscillating mass: 𝓍(t) = (2.0 cm) cos(10t - π/4). Here, the amplitude is 2.0 cm, the angular frequency is 10 rad/s, and the phase constant is -π/4. The initial conditions refer to the position and velocity of the mass at t = 0.

Step 2: To find the initial position, substitute t = 0 into the position equation. This gives 𝓍(0) = (2.0 cm) cos(-π/4). Use the trigonometric identity cos(-θ) = cos(θ) to simplify the expression.

Step 3: To find the initial velocity, differentiate the position equation with respect to time to get the velocity equation. The derivative of 𝓍(t) = (2.0 cm) cos(10t - π/4) is v(t) = -Aω sin(ωt - φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. Substituting the given values, v(t) = -(2.0 cm)(10 rad/s) sin(10t - π/4).

Step 4: Substitute t = 0 into the velocity equation to find the initial velocity. This gives v(0) = -(2.0 cm)(10 rad/s) sin(-π/4). Use the trigonometric identity sin(-θ) = -sin(θ) to simplify the expression.

Step 5: Combine the results from Step 2 and Step 4 to summarize the initial conditions: the initial position 𝓍(0) and the initial velocity v(0). These values describe the state of the oscillating mass at t = 0.

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주요 개념

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

Oscillation

Oscillation refers to the repetitive variation, typically in time, of some measure about a central value or between two or more different states. In this context, it describes the motion of the mass as it moves back and forth around an equilibrium position, characterized by parameters such as amplitude, frequency, and phase.

Initial Conditions

Initial conditions are the values of the variables at the start of the observation or experiment, which in this case refers to the position and velocity of the oscillating mass at time t = 0. These conditions are crucial for determining the future behavior of the system and are derived from the given position function.

Phase Angle

The phase angle in oscillatory motion indicates the initial position of the oscillating object relative to a reference point. In the equation provided, the phase angle of -π/4 affects the starting position of the mass, shifting the cosine function and thus altering the initial conditions of the oscillation.