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. Start by identifying the type of motion involved. The block attached to the spring is undergoing simple harmonic motion (SHM). The key parameters for SHM are mass (m), spring constant (k), initial velocity, and initial position.
2. Use the formula for the angular frequency \( \omega \) of a spring-mass system: \( \omega = \sqrt{\frac{k}{m}} \). Substitute the given values: \( k = 300 \text{ N/m} \) and \( m = 2.00 \text{ kg} \) to find \( \omega \).
3. To find the amplitude (A), use the energy conservation principle. The total mechanical energy in SHM is given by \( E = \frac{1}{2} k A^2 \). At \( t = 0 \), the kinetic energy is \( \frac{1}{2} m v^2 \). Set the kinetic energy equal to the total energy to solve for \( A \).
4. For the phase angle (\( \phi \)), use the initial conditions. The position \( x(t) \) in SHM is given by \( x(t) = A \cos(\omega t + \phi) \). At \( t = 0 \), the spring is neither stretched nor compressed, so \( x(0) = 0 \). Use the initial velocity \( v(0) = -12.0 \text{ m/s} \) and the velocity equation \( v(t) = -A \omega \sin(\omega t + \phi) \) to solve for \( \phi \).
5. Write the equation for the position as a function of time using the amplitude and phase angle found: \( x(t) = A \cos(\omega t + \phi) \). Substitute the values of \( A \), \( \omega \), and \( \phi \) to express \( x(t) \) in terms of known quantities.

Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes the oscillatory motion of systems like springs and pendulums, where the restoring force is proportional to the displacement. In this problem, the block attached to the spring exhibits SHM, characterized by sinusoidal functions of time for displacement, velocity, and acceleration.

Amplitude of Oscillation

The amplitude in SHM is the maximum displacement from the equilibrium position. It represents the energy stored in the system and is determined by initial conditions such as velocity and position. For this problem, the amplitude can be calculated using the initial velocity and the properties of the spring.

Phase Angle in SHM

The phase angle in SHM determines the initial position and direction of motion at t = 0. It is crucial for writing the equation of motion, as it shifts the sine or cosine function to match the initial conditions. Calculating the phase angle involves using the initial velocity and position of the block.