91Ó°ÊÓ

Spring force is a fundamental concept in understanding how springs work in mechanical systems. When a spring is either compressed or stretched from its natural length, it exerts a force to return to its equilibrium position. This force follows Hooke's Law, which states that the force (F) exerted by a spring is proportional to the displacement (x) from its equilibrium position: \[ F = -kx \]Where \( k \)is the spring constant, representing the stiffness of the spring. A higher \( k \)means a stiffer spring. In the NASA test vehicle scenario, the spring stretches due to the force exerted by the ball's mass, influenced by the vehicle's acceleration. This pseudoforce acts as if it's pulling the spring, causing it to elongate until the engines are turned off, at which point the spring begins its oscillation.

The amplitude of oscillation is the maximum displacement of the system from its equilibrium position. In this problem, it can be calculated by considering the initial stretch of the spring due to pseudoforce. Pseudoforce arises in non-inertial reference frames, such as within an accelerating vehicle. Since the vehicle accelerates at 5 m/s², the pseudoforce can be calculated using \( F = ma \), yielding \( 17.5 \text{ N} \).
The spring elongates according to Hooke's Law \( F = kx \), giving \( x = \frac{F}{k} = 0.078 \text{ meters} \). This initial stretch is the amplitude, as it's the maximum distance the spring reaches from its equilibrium position once oscillation starts.

The frequency of oscillation determines how fast the oscillations occur in a system. For a mass-spring system like the one in this exercise, the natural frequency can be calculated using the formula:\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]This indicates that the frequency depends on the spring constant \( k \)and the mass \( m \)of the ball. A higher \( k \)or a smaller mass \( m \) increases the frequency, meaning more oscillations per second. For the given \( k = 225 \text{ N/m} \) and \( m = 3.5 \text{ kg} \), calculation yields a frequency of approximately 1.276 Hz. Thus, the ball goes through a little more than one cycle per second.

Pseudoforce, also known as fictitious or inertial force, is a force that seems to act on a mass during acceleration when observed from a non-inertial reference frame, like inside an accelerating vehicle. Unlike real forces, pseudoforce arises purely from the acceleration of the reference frame itself.

• For a car accelerating forward, pseudoforce acts backward on the occupants or objects inside.
• It is proportional to the acceleration of the vehicle and the mass of the object inside.

In our situation, while the NASA vehicle was accelerating, a pseudoforce of \( 17.5 \text{ N} \)stretched the spring from its equilibrium. Once the acceleration stopped with the engines turning off, the pseudoforce disappeared, allowing the spring to oscillate freely from its stretched position.

To find the maximum speed of a mass in a harmonic oscillator, use the formula:\[ v_{max} = A\omega \]Here, \( A \)represents the amplitude, and \( \omega \)is the angular frequency. Angular frequency is related to the oscillation frequency by:\[ \omega = 2\pi f \]For the calculated amplitude (\( A = 0.078 \text{ m} \)) and frequency (\( f = 1.276 \text{ Hz} \)), \( \omega \)turns out to be about \( 8.02 \text{ rad/s} \).
The maximum speed of the ball relative to the vehicle is then calculated to be roughly \( 0.626 \text{ m/s} \),which is the peak speed it reaches as the spring moves back through its equilibrium position during oscillation.