A 0.500-kg mass on a spring has velocity as a function of time given by vx(t) = -(3.60 cm/s) sin[(4.71 rad/s)t - (pi/2)]. What are (a) the period; (b) the amplitude; (c) the maximum acceleration of the mass; (d) the force constant of the spring?

Verified step by step guidance

1. Identify the angular frequency (\(\omega\)) from the velocity function \(v_x(t) = -(3.60 \text{ cm/s}) \sin[(4.71 \text{ rad/s})t - (\pi/2)]\). The angular frequency \(\omega\) is the coefficient of \(t\) in the sine function, which is 4.71 rad/s.
2. Calculate the period (\(T\)) of the motion using the relationship between period and angular frequency: \(T = \frac{2\pi}{\omega}\).
3. Determine the amplitude of the velocity (\(A_v\)), which is the coefficient in front of the sine function in the velocity equation. Convert this amplitude from cm/s to m/s for consistency in units.
4. Find the maximum acceleration (\(a_{\text{max}}\)) using the relationship between maximum acceleration and maximum velocity in simple harmonic motion: \(a_{\text{max}} = \omega^2 \times A_x\), where \(A_x\) is the amplitude of the displacement. Use the relationship between the amplitudes of velocity and displacement in simple harmonic motion: \(A_v = \omega \times A_x\).
5. Calculate the force constant (\(k\)) of the spring using Hooke's Law and the relationship between the force constant, mass, and angular frequency: \(k = m \omega^2\).

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. In this case, the mass on the spring exhibits SHM, characterized by sinusoidal functions for position and velocity. Understanding SHM is crucial for analyzing the motion of the mass and deriving properties like period and amplitude.

Period and Frequency

The period of a motion is the time taken for one complete cycle of oscillation, while frequency is the number of cycles per unit time. For SHM, the period (T) is related to the angular frequency (ω) by the formula T = 2π/ω. In the given equation, the angular frequency can be directly identified, allowing for the calculation of the period of the mass-spring system.

Maximum Acceleration

The maximum acceleration in SHM occurs at the maximum displacement and is given by the formula a_max = ω²A, where A is the amplitude and ω is the angular frequency. This concept is essential for determining how quickly the mass changes its velocity at the extremes of its motion. Understanding this relationship helps in calculating the maximum acceleration of the mass in the given problem.