Question

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y = 10 e^{- \frac{t}{2}} \cos (\frac{π t}{8})$ .
a. Graph the displacement function.

Accepted Answer

Identify the components of the displacement function: The function is y = $$10e^{-t/2} * cos$$(πt/8). It consists of an exponential decay term $$10e^{-t/2}$$ and a cosine oscillation term cos(πt/8). Understand the effect of each component: The exponential term $$10e^{-t/2}$$ represents damping, which means the amplitude of the oscillation decreases over time. The cosine term cos(πt/8) represents the oscillatory motion of the mass on the spring. Determine the behavior of the function over time: As time t increases, the exponential term decreases, causing the overall amplitude of the oscillation to decrease. The cosine term will continue to oscillate between -1 and 1, but with a decreasing amplitude due to the damping effect. Set up a graphing tool or software: Use a graphing calculator or software capable of plotting functions to visualize the displacement function. Input the function y = $$10e^{-t/2} * cos$$(πt/8) into the tool. Plot the function: Observe the graph to see how the displacement changes over time. Notice the oscillations with decreasing amplitude, which is characteristic of a damped oscillator. The graph should show a wave-like pattern that diminishes as time progresses.

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y = 10 e^{- \frac{t}{2}} \cos (\frac{π t}{8})$ .
a. Graph the displacement function.

Verified video answer for a similar problem:

Key Concepts

Damped Oscillator

Exponential Function

Cosine Function

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by y=10e−t2cos(πt8)y=10e^{-\(\frac{t}{2}\)}\(\cos\[\left\)(\(\frac{\pi t}{8}\]\right\)).a. Graph the displacement function.

Verified video answer for a similar problem:

Key Concepts

Damped Oscillator

Exponential Function

Cosine Function

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y = 10 e^{- \frac{t}{2}} \cos (\frac{π t}{8})$ .
a. Graph the displacement function.