91Ó°ÊÓ

In the context of Simple Harmonic Motion (SHM), the amplitude represents the maximum extent of displacement from the equilibrium position. Think of it as the furthest point the object reaches from the center, whether that’s to the left or right. In this exercise, the amplitude is given as 0.240 meters. This means the bolt moves 0.240 meters above and below the center position during its motion.

Understanding amplitude is crucial because it determines the range of movement within the motion cycle. An object with a larger amplitude will move farther from the equilibrium, which can also indicate higher energy in the system.

• Amplitude is always a positive value and is expressed in meters in SHM.
• It helps us predict how far an object will go in its oscillatory path.

Angular frequency is a term associated with how fast something oscillates in SHM. It's denoted by the symbol \( \omega \) and ties directly to the period \( T \), or the time it takes to complete one full cycle of motion.

Mathematically, angular frequency is calculated as \( \omega = \frac{2\pi}{T} \). So, when you know the period, you can find the angular frequency, which in this exercise is \( \frac{4\pi}{3} \) radians per second.
To better grasp this:

• Angular frequency lets us understand the relationship between time intervals and rotational angles.
• It is measured in radians per second, not to be confused with regular frequency which is cycles per second or Hertz (Hz).

In essence, a higher angular frequency implies quicker oscillations, meaning the object cycles back and forth more times within a given time frame.

Displacement in SHM refers to the current position of the moving object relative to its equilibrium position. It's a dynamic value, changing continuously as the object oscillates back and forth.

The displacement can be positive, negative, or zero. A positive displacement indicates the object is on one side of the equilibrium position, while a negative value specifies the opposite side. To determine displacement at a particular time, you use the equation \( x(t) = A \cos(\omega t + \phi) \), which connects amplitude, angular frequency, and phase constant.
In our example, when \( t = 0.500 \) seconds, the bolt’s displacement calculated using the formula turns out to be -0.120 meters.

• Displacement is critical for determining other properties, like velocity and acceleration.
• It provides insight into the position of the object at any time during the motion.

The spring constant, often symbolized as \( k \), is a property that measures the stiffness of a spring in SHM. It tells us how much force is required to stretch or compress the spring by a unit length. The unit for the spring constant is Newton per meter (N/m).

In problems involving springs, like with our bolt in SHM, the spring constant can be found using the relation \( k = m\omega^2 \), where \( m \) is the mass of the object, and \( \omega \) is the angular frequency. For the bolt, we found \( k \) to be approximately 0.1121 N/m.
Understanding the spring constant helps us predict the system’s behavior under different conditions:

• A higher spring constant means the spring is stiffer, and more force is needed to produce the same displacement.
• Helps us calculate other forces and predict the motion in systems involving springs.