The equation for an ellipse is \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1 .\) Show that two-dimensional simple harmonic motion whose components have different amplitudes and are \(\pi / 2\) out of phase gives rise to elliptical motion. How are constants \(a\) and \(b\) related to the amplitudes?

An ellipse, a fundamental shape in physics and geometry, can be visualized as a stretched circle where two points, called foci, replace the single central point. Described mathematically, the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) captures the essence of an ellipse. Here, \(a\) and \(b\) represent the semi-major and semi-minor axes that dictate the ellipse's shape and size.

In the context of simple harmonic motion (SHM), we often come across elliptical paths when analyzing movements in two dimensions. When the motion in the x and y axes are both simple harmonic and there is a phase difference, specifically \(\pi/2\), the resultant motion draws an ellipse. To visualize this, imagine two springs oscillating at right angles to each other; one controls the x-direction while the other controls the y-direction. If their oscillations are coordinated correctly, the combined path traces out an ellipse.

The constants \(a\) and \(b\) in the elliptical equation are directly related to the amplitudes of the oscillations in the x and y directions, respectively. These amplitudes determine how 'wide' or 'tall' the ellipse will be, thereby forming the physical bridge between the abstract equation and the tangible path of motion in SHM.

Amplitude, often denoted as \(A\) in physics, is a crucial concept in understanding simple harmonic motion. It represents the maximum extent of the oscillation or the furthest distance from the equilibrium position that the object moves.

When considering SHM, it's helpful to think of a child on a swing or a mass bobbing on a spring. The amplitude is the highest height the child achieves on the swing or the maximum stretch or compression of the spring. This concept applies to two-dimensional SHM as well; the separate amplitudes along the x-axis and y-axis define the extremities of the motion in each dimension.

Returning to our elliptical motion, the amplitude in the x-direction, \(A\), and the amplitude in the y-direction, \(B\), directly determine the size of the ellipse—specifically, the lengths of its semi-major and semi-minor axes. Therefore, understanding amplitude is key to predicting and describing the physical displacement in SHM.

In SHM, the term 'phase difference' refers to the angular difference by which one oscillatory motion leads or lags another. Think of two dancers performing the same routine but starting at different times—similarly, in SHM, two waves or motions can have the same frequency but be out of step with each other.

In the exercise, the phase difference between the x and y components is specified as \(\pi/2\) radians, or 90 degrees, indicating quarter of a cycle phase shift. This difference is what leads to the x and y motions to be perpendicular in phase, resulting in an elliptical path when one motion is at its maximum, the other is at zero, and vice versa.

This concept of phase difference is not only crucial for understanding the elliptical motion in SHM but also plays a significant role in various fields of physics, such as wave interference, alternating current circuits, and even the beating of heart muscles. It's important for students to grasp the influence of phase difference, as it often determines the constructive or destructive interference between waves and the overall pattern of motion in oscillatory systems.