Question

The equation \( 3 \frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 15x = 20 \cos 4t \) shows a vibrations system. The maximum amplitude of the system will be:

• A. 50 cm
• B. 51 cm
• C. 49 cm
• D. 52 cm

Hint:

When dealing with oscillating systems, the maximum amplitude can be found by applying resonance and damping principles. Ensure to correctly substitute values for mass, damping coefficient, and frequency.

Answer 1

The Correct Option is B

This equation represents a second-order linear differential equation for a damped harmonic oscillator. The solution to such a system typically involves finding the resonance condition and calculating the steady-state amplitude. To calculate the maximum amplitude of the system, we use the formula for forced vibrations: \[ X_{max} = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (c\omega)^2}} \] where \( F_0 \) is the amplitude of the forcing function, \( k \) is the spring constant, \( m \) is the mass, \( c \) is the damping coefficient, and \( \omega \) is the frequency. From the equation, the system's damping coefficient and frequency are determined by the constants in the equation. Solving for the maximum amplitude gives approximately 51 cm.