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 velocity at t = 0.40 s.

검증된 단계별 안내

Step 1: Recall the relationship between position and velocity in simple harmonic motion. Velocity is the time derivative of the position function. Start by differentiating the given position function 𝓍(t) = (2.0 cm) cos(10 t ─ π/4) with respect to time t.

Step 2: Apply the derivative rule for cosine: d/dt[cos(θ)] = -sin(θ) * dθ/dt. Here, θ = (10 t ─ π/4), so dθ/dt = 10. The derivative of 𝓍(t) becomes v(t) = - (2.0 cm) * sin(10 t ─ π/4) * 10.

Step 3: Simplify the expression for velocity: v(t) = -20 cm/s * sin(10 t ─ π/4). This is the general formula for the velocity of the oscillating mass.

Step 4: Substitute t = 0.40 s into the velocity equation. Calculate the argument of the sine function: θ = 10 * 0.40 ─ π/4.

Step 5: Evaluate sin(θ) using the calculated θ value, and multiply by -20 cm/s to find the velocity at t = 0.40 s. Ensure the units are consistent throughout the calculation.

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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 can be described by sinusoidal functions, such as sine or cosine, which represent the position of the object as a function of time. In this case, the mass oscillates with a specific amplitude and angular frequency, which are key to determining its velocity and other properties.

Velocity in SHM

In Simple Harmonic Motion, the velocity of an oscillating object can be derived from its position function. The velocity is the time derivative of the position function, indicating how fast the object is moving and in which direction. For the given position function, differentiating it with respect to time will yield the velocity expression, which can then be evaluated at any specific time.

Angular Frequency

Angular frequency, denoted by ω, is a measure of how quickly an object oscillates in radians per second. It is related to the frequency of oscillation and is crucial for determining the motion characteristics of oscillating systems. In the provided equation, the angular frequency is represented by the coefficient of t in the cosine function, which influences both the speed and the period of the oscillation.