91Ó°ÊÓ

Angular frequency, often symbolized as \( \omega \), is a key element in simple harmonic motion (SHM) and it tells us how quickly an object oscillates. In the context of the given exercise, the angular frequency is positioned as the multiplier of time \( t \) in the cosine function of the displacement equation. Specifically, it is \( 4.16 \, \text{s}^{-1} \).

This constant provides insight into how many radians the oscillation completes per second. To find the time it takes to complete one full cycle, we use the period \( T \) formula: \[T = \frac{2\pi}{\omega}\] Plugging in \( \omega = 4.16 \, \text{s}^{-1} \), the period is computed as approximately 1.51 seconds. This means that in this SHM scenario, the mass completes one full oscillation in about 1.51 seconds.

The spring constant, represented as \( k \), signifies the stiffness of a spring in simple harmonic motion. It is determined from the relationship between angular frequency and mass of the oscillating object. The equation linking these parameters is:

• \( \omega = \sqrt{\frac{k}{m}} \)

Rearranging this for \( k \) gives us:\[k = m\omega^2\]Given \( m = 1.50 \, \text{kg} \) and \( \omega = 4.16 \, \text{s}^{-1} \), the spring constant is calculated as \( k = 1.50 \times (4.16)^2 \approx 26.0 \, \text{N/m} \).

This reveals that the spring requires 26.0 Newtons of force to stretch it by one meter, indicating how rigid or flexible the spring is during oscillations.

In simple harmonic motion, the maximum speed \( v_{\text{max}} \) of the oscillating object is a crucial aspect to grasp, as it relates to how quickly the object moves. This speed is achieved when the object's displacement is zero. The formula for maximum speed is:

• \( v_{\text{max}} = A \omega \)

where \( A \) is the amplitude of oscillation and \( \omega \) is the angular frequency.

For the exercise at hand, with \( A = 0.074 \, \text{m} \) (converted from 7.40 cm) and \( \omega = 4.16 \, \text{s}^{-1} \), the maximum speed is \( v_{\text{max}} = 0.074 \times 4.16 \approx 0.308 \, \text{m/s} \).

This means that the mass achieves its quickest motion speed of 0.308 meters per second as it passes through the equilibrium position in its oscillatory path.

The displacement function describes how the position of an object in simple harmonic motion varies over time. In our specific equation, it is:\[x(t) = (7.40 \, \text{cm}) \cos[(4.16 \, \text{s}^{-1})t - 2.42]\]This function uses cosine to show the oscillation profile, where:

• The amplitude \( A = 7.40 \, \text{cm} \) indicates the maximum distance from the equilibrium.
• The term \( 4.16 \, \text{s}^{-1} \) corresponds to the angular frequency.
• \(-2.42\) is the phase shift altering the starting point of the oscillation.

Plugging in a particular time, like \( t=1.00 \, \text{s} \), gives us the precise position at that moment, \( x(1.00) \approx 0.065 \, \text{m}\).

This demonstrates how the displacement at any point in time can be determined, revealing the mass's precise location as it swings back and forth.