Hello Futurestar33. Welcome to PF !Futurestar33 said:Homework Statement
I am curious as to how the second line in the equation is equal to the third line in the equation. The book my class is using is Taylor and it just skips so many steps. What happens to the sign, I know this must relate to euler in some way I am just not sure how. Thank you
I am just curious as to why and how
Homework Equations
The Attempt at a Solution
Harmonic motion in one dimension refers to the back-and-forth motion of an object along a straight line, where the object's acceleration is directly proportional to its displacement from a central equilibrium point. This type of motion follows a sinusoidal pattern and is commonly seen in pendulums, springs, and other oscillating systems.
The equation for harmonic motion in one dimension is x = A cos(ωt + φ), where x is the displacement from the equilibrium point, A is the amplitude of the motion, ω is the angular frequency (related to the period of the motion), and φ is the phase angle.
The equation for harmonic motion can be derived using Newton's second law of motion, which states that the sum of forces acting on an object is equal to its mass times its acceleration. By considering the forces acting on an object in harmonic motion (such as gravity and spring force), we can derive the equation x = A cos(ωt + φ) as a solution to the differential equation of motion.
The amplitude in harmonic motion represents the maximum displacement of the object from its equilibrium point. It is a measure of the object's energy and determines the height and width of the oscillations. The larger the amplitude, the more energy the system has, and the farther it will move from the equilibrium point.
The frequency of harmonic motion, which is related to the angular frequency ω, determines how quickly the object oscillates back and forth. A higher frequency means the object completes more cycles in a given time, resulting in faster motion. The frequency is also inversely proportional to the period of the motion, so a higher frequency corresponds to a shorter period.