91Ó°ÊÓ

The angular coordinate, often denoted by \( \theta \), is a measure of the rotational position of an object in radians. In this case, the formula given describes the oscillating flywheel's angular position as a function of time \( t \): \[ \theta = \theta_{0} e^{-7 \pi v t} \sin 4 \pi t \]

• \( \theta_{0} \) is the initial angular displacement, measured in radians, here provided as 0.4 rad.
• The term \( e^{-7 \pi v t} \) represents an exponential decay, which means the influence of initial conditions reduces over time.
• The \( \sin 4 \pi t \) component describes the oscillatory nature of the flywheel, introducing periodic motion.

To determine the angular coordinate at a specific time, you substitute the values into the equation. For example, at \( t = 0.125 \, \mathrm{s} \), you would plug in these values, compute the sine component, and then calculate \( \theta \). Over time, as \( t \rightarrow \infty \), \( \theta(t) \) approaches zero due to the exponent dominating.

Angular velocity, represented as \( \omega(t) \), describes how fast the angular position changes with time. It is calculated by differentiating the angular coordinate \( \theta(t) \) with respect to time \( t \). For this exercise, we utilize the product rule to differentiate: \[ \omega(t) = \frac{d}{dt}(0.4 e^{-7 \pi v t} \sin 4 \pi t) \] The differentiation yields:

• \( -2.8 \pi v e^{-7 \pi v t} \sin(4 \pi t) \), which accounts for the decay in amplitude due to the derivative of the exponential term.
• \( +1.6 \pi e^{-7 \pi v t} \cos(4 \pi t) \), due to the derivative of the sine term introducing a cosine component.

This results in a more complex expression than \( \theta(t) \), combining both sine and cosine influences. At \( t = 0.125 \, \mathrm{s} \), you substitute this to find \( \omega \). As time approaches infinity, the influence of the decay term ensures \( \omega(t) \) trends toward zero.

Angular acceleration, denoted by \( \alpha(t) \), refers to the rate of change of angular velocity. We achieve this by differentiating the previously obtained angular velocity expression with respect to time \( t \). Again, using the product and chain rules, we continue to break down each component from the formula: There are multiple parts involved:

• The decay term \( e^{-7 \pi v t} \) is differentiated again, affecting both sine and cosine contributions.
• The sine and cosine components are differentiated, leading to secondary terms and further adjustments in the angular motion description.

The resulting expression can be intricate, involving negative and positive contributions due to exponential decreasing factors and oscillatory behavior. Evaluating \( \alpha(t) \) at \( t = 0.125 \, \mathrm{s} \) involves plugging in the numeric value and calculating accordingly. As with \( \theta(t) \) and \( \omega(t) \), \( \alpha(t) \) decays to zero when \( t \rightarrow \infty \), reflecting the diminishing dynamic activity of the oscillating flywheel.