A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At t = 0 s, the mass is at x = 5.0 cm and has vₓ = ─30 cm/s. Determine: g. The total energy.

Verified step by step guidance

1. Identify the given values: mass (m) = 200 g = 0.2 kg, frequency (f) = 2.0 Hz, initial position (x₀) = 5.0 cm = 0.05 m, and initial velocity (vₓ₀) = -30 cm/s = -0.3 m/s.
2. Calculate the angular frequency ($\omega$) using the formula $\omega =2\pi f$.
3. Use the formula for the potential energy stored in the spring at the initial position, which is $U=\frac{1}{2}k{x}_0^2$. Here, $k$ is the spring constant, which can be found using $k=m{\omega }^2$.
4. Calculate the kinetic energy (K) at the initial position using $K=\frac{1}{2}m{v}_0^2$.
5. Add the potential energy and kinetic energy to find the total mechanical energy (E) of the system: $E=U+K$.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In SHM, the restoring force is directly proportional to the displacement from the equilibrium and acts in the opposite direction. The motion can be described by sinusoidal functions, and key parameters include amplitude, frequency, and phase. Understanding SHM is essential for analyzing systems like springs and pendulums.

Mechanical Energy in Oscillatory Systems

In oscillatory systems, mechanical energy is conserved and consists of potential energy stored in the spring and kinetic energy of the mass. The total mechanical energy (E) can be expressed as E = 1/2 k A², where k is the spring constant and A is the amplitude of oscillation. At any point in the motion, the sum of kinetic and potential energy remains constant, allowing for the calculation of total energy at different positions.

Frequency and Angular Frequency

Frequency (f) is the number of oscillations per unit time, measured in Hertz (Hz), while angular frequency (ω) relates to the frequency by the equation ω = 2πf. Angular frequency is crucial for understanding the dynamics of oscillatory motion, as it appears in equations describing the position and velocity of the mass in SHM. In this problem, knowing the frequency allows for the calculation of other parameters like the spring constant and total energy.