A 2.00-kg, frictionless block is attached to an ideal spring with force constant 300 N/m. At t = 0 the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 m/s. Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

Verified step by step guidance

1. Identify the type of motion: The problem describes a mass-spring system undergoing simple harmonic motion (SHM) since the block is attached to an ideal spring and there is no friction.
2. Determine the amplitude (A): The amplitude in SHM is the maximum displacement from the equilibrium position. Use the formula for the maximum speed in SHM, v_max = A \omega, where \omega is the angular frequency. Rearrange to solve for A: A = v_max / \omega. Calculate \omega using \omega = \sqrt{k/m}, where k is the spring constant and m is the mass.
3. Find the phase angle (\phi): At t = 0, the spring is at its natural length and the block is moving in the negative direction. This indicates that the motion is at its maximum velocity in the negative direction. Use the sine function for velocity, v(t) = -A\omega \sin(\phi), and solve for \phi when v(t) = -12.0 m/s and t = 0.
4. Write the position function x(t): The general form for the position as a function of time in SHM is x(t) = A \cos(\omega t + \phi). Substitute the values of A, \omega, and \phi obtained from the previous steps.
5. Verify units and signs: Ensure that all units are consistent (e.g., meters, seconds) and check the signs, especially for the phase angle, to ensure they correspond to the described motion direction.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In this case, the block attached to the spring will move back and forth due to the restoring force exerted by the spring, which is proportional to its displacement from the equilibrium position. The characteristics of SHM include amplitude, frequency, and phase, which are essential for analyzing the motion.

Amplitude

Amplitude is the maximum extent of displacement from the equilibrium position in oscillatory motion. It represents the maximum distance the block moves from its rest position when the spring is either fully compressed or stretched. In this problem, the amplitude can be determined using the initial conditions of the block's motion and its energy conservation principles.

Phase Angle

The phase angle in SHM describes the initial position and direction of motion of the oscillating object at time t=0. It is crucial for determining the specific form of the position function over time. The phase angle can be calculated based on the initial velocity and position of the block, allowing us to write the complete equation for its motion.