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 angular frequency: \(a_{\text{max}} = \omega^2 \times A_x\), where \(A_x\) is the amplitude of the displacement. Use the relationship between the amplitude of velocity and displacement: \(A_x = \frac{A_v}{\omega}\).
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 context, the mass on the spring exhibits SHM, characterized by sinusoidal functions for position and velocity, which can be analyzed to determine key parameters 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) can be derived from the angular frequency (ω) using the relationship T = 2π/ω. In this case, the angular frequency is given as 4.71 rad/s, allowing for the calculation of the period.

Maximum Acceleration and Force Constant

The maximum acceleration in SHM is given by the formula a_max = ω²A, where A is the amplitude of the motion. The force constant (k) of the spring relates to the mass (m) and angular frequency (ω) through the equation k = mω². Understanding these relationships is crucial for determining the maximum acceleration and the spring's force constant from the given parameters.