91Ó°ÊÓ

A vibrating particle is an object moving in a repeated back-and-forth motion due to an applied force or its inherent dynamic properties. In the context of the exercise, the particle's motion is depicted by the position vector \(\mathbf{r} = (4 \sin \pi t) \mathbf{i} - (\cos 2 \pi t) \mathbf{j}\). This includes sinusoidal elements that describe how the particle moves with respect to time \(t\) in both the \(x\) and \(y\) directions.
The sinusoidal functions represent periodic motion, a fundamental trait of vibrating systems. In this scenario:

• The \(4 \sin \pi t\) component indicates the amplitude and frequency of vibration along the x-axis.
• The \(-\cos 2 \pi t\) component offers a similar description for the y-axis. Notice that the y-axis part involves a cosine function, suggesting a phase shift from the sine wave representing x-motion.

Understanding these components helps recognize how the position changes over time, depicting the typical oscillation pattern of a vibrating particle.

Velocity and acceleration are key parameters in understanding the motion of particles. Velocity refers to how fast an object is moving and in what direction, while acceleration measures the change in velocity over time.
For the problem at hand, the velocity vector \(\mathbf{v}\) is the derivative of the position vector with respect to time \(t\), resulting in:

• \(\mathbf{v} = 4 \pi \cos \pi t \mathbf{i} + 2 \pi \sin 2 \pi t \mathbf{j}\)

Acceleration, being the derivative of velocity, comes next:

• \(\mathbf{a} = -4 \pi^2 \sin \pi t \mathbf{i} + 4 \pi^2 \cos 2 \pi t \mathbf{j}\)

When you substitute \(t = 1\) second, the velocity simplifies to \(-4 \pi \mathbf{i}\) and acceleration to \(4 \pi^2 \mathbf{j}\). This showcases how these calculations reveal the actual speed and direction of a particle, essential for predicting future motion.

A parabolic path occurs when the trajectory of an object creates a curve that follows the shape of a parabola. In the provided exercise, the goal is to show that the path of the vibrating particle is indeed parabolic.
This is accomplished through parameter elimination within the position equations \(x(t) = 4 \sin \pi t\) and \(y(t) = -\cos 2 \pi t\). Using trigonometric identities and algebraic manipulation:

• You relate \(\cos 2 \pi t\) to the \(x\) coordinate.
• Solve for \(\sin \pi t\) in terms of \(x\), proving that \((x/4) = \sin \pi t\).

This substitution shows that the relationship between \(x\) and \(y\) follows a parabolic equation, without any direct reference to time \(t\). Recognizing this path supports the understanding of particle trajectories in kinematics.

Kinematics is the branch of physics that studies the motion of objects without considering the forces causing them. It focuses on position, velocity, and acceleration.
In classical mechanics, kinematic equations offer insights into motion in various dimensions. For a vibrating particle moving in a 2D plane, kinematics facilitates the understanding of how position and velocity change over time.
This exercise demonstrates:

• The clear calculation of derivative functions to obtain velocity and acceleration.
• The application of trigonometric identities to describe motion paths, like parabolas.

Studying kinematics allows one to predict and analyze trajectories and motion properties, crucial for engineering, physics problem-solving, and everyday phenomena.