Question

A particle of mass \(m\) moves in a fixed plane along the trajectory \(\overrightarrow{\mathbf{r}}=\hat{\mathbf{i}} A \cos \omega t+\hat{\mathbf{j}} A \cos 3 \omega t .\) (a) Sketch the trajectory of the particle. (b) Find the force acting on the particle. Also find (c) its potential energy and \((d\) ) its total energy as functions of time. \((e)\) Is the motion periodic? If so, find the period.

Short answer

The trajectory will be in the form of a Lissajous figure. The force acting on a particle can be found by differentiating the position function twice and the potential, kinetic, total energy functions can be calculated based on their definitions. The motion is periodic with a period of \(2\pi / \omega\).

Step-by-step solution

Sketch the Trajectory

The trajectory represents the path that an object follows. The given position vector is \(\overrightarrow{\mathbf{r}}=\hat{\mathbf{i}} A \cos \omega t+\hat{\mathbf{j}} A \cos 3 \omega t\). Here \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) are unit vectors along the x and y-axes. The x and y coordinates of the particle are varying with respect to time and they show a harmonic oscillation with frequencies \(\omega\) (for x) and \(3\omega\) (for y). To sketch the trajectory, create a graph with x and y being independent variables, then plot the given cosinusoidal functions.

Find the Force Acting on the Particle

Force \(F\) on a particle of mass \(m\) is given by Newton's second law \(F = m.a\), where \(a\) is acceleration. For the provided position function, find the acceleration by differentiating the position function twice with respect to time \(t\). Then, multiply it by mass \(m\) to get the force.

Find from the Potential Energy

Potential energy \(U\) in mechanics is dependent on position, and is connected to the force acting on the particle by \(F = -\nabla U\). For simplicity, assume that any effective potential has the form \(U = U(x)\). To find the potential energy, integrate the obtained force function with respect to \(x\), and add an arbitrary constant of integration \(C\).

Calculate the Total Energy

The total energy \(E\) of the system is the sum of the kinetic energy \(K = \frac{1}{2} m v^2\) and the potential energy \(U\). Kinetic energy can be found by differentiating the given position function once to get the velocity, and then plugging that into the formula for kinetic energy. Then add the previously calculated potential energy function to find the total energy function of time.

Determine the Periodicity of Motion

The motion is said to be periodic if it repeats at regular intervals. In this scenario, observe the functions of \(x\) and \(y\) coordinates. These coordinates show harmonic motion and repeat their value after every \(2\pi / \omega\) and \(2\pi / 3\omega\). Find the least common multiple of these two periods, which would be the period of the motion.

Key concepts

Trajectory Sketching

Imagine a particle zipping around in space; trajectory sketching is much like drawing out its exact path. In our exercise, the particle moves harmonically along a plane, following the equations \(x(t) = A \cos(\omega t)\) and \(y(t) = A \cos(3\omega t)\), where \(A\) and \(\omega\) are constants that control the amplitude and frequency of the motion, respectively.

To visualize this path, we draw two cosinusoidal waves on a graph—one for the x-coordinate and another for the y-coordinate. As time t advances, the particle goes back and forth, tracing out a unique shape, which could resemble a Lissajous curve, indicating that its motion is confined to a fixed plane. By methodically plotting these equations against time, we can create a trajectory sketch that is both accurate and insightful into the particle's movements.

Newton's Second Law

At the heart of many physics problems, including the motion of our particle, lies Newton's second law. Formally stated as \(F = ma\), it's a straightforward yet profound way of saying, 'to accelerate an object, you must push it.' In simpler terms, force is the product of mass and acceleration—a cornerstone of classical mechanics. When we calculate the particle's force as in Step 2, we're effectively figuring out the necessary 'push' at every moment in time as it jogs along its path. To deduce this force, we accelerate to the land of calculus and differentiate the position function twice, then multiply by the mass; the resultant expression represents the exact magnitude and direction of the force required to keep the particle in such a harmoniously oscillating state.

Potential Energy in Mechanics

Potential energy in mechanics is essentially 'stored energy.' Think of it like a bank account of energy, waiting to be withdrawn when the particle moves about its path. In our exercise, since we're dealing with a conservative force, potential energy is tied to the particle's position along the trajectory. As the particle ascends the graph, it's as though it's climbing a hill, storing energy that can later be transformed into kinetic energy as it rolls down. We find this potential energy by playing a game of opposites with the force—integrating it with respect to its position. It's a delicate dance of classical physics—every push from a force corresponds to a hike up the potential energy hill, and the force we're seeking is like the downhill slope at any point along the way.

Total Energy of a System

Total energy—the sum of all energies—is the grand tally of kinetic energy (energy of motion) and potential energy (stored energy). It’s an unchanging figure for a closed system, thanks to the conservation of energy principle. For our harmoniously moving particle, we calculated kinetic energy—\(\frac{1}{2} m v^2\)—by finding the velocity (with a dash of calculus magic) and then squared it for good measure. Add that to the potential energy, and voilà, you've arrived at total energy. It's a momentous sum that encapsulates the full tale of the particle's mechanical energy throughout its oscillatory adventures.

Periodicity of Motion

Periodicity is the heartbeat of motion in physics—the consistent throb of actions recurring over time. A periodic motion is as dependable as Old Faithful, repeating its performance after a set interval. For our particle, by examining the functions for x and y, we can spot this repetition: every time \(\omega t\) advances by \(2\pi\), the functions go through one full cycle. To pinpoint the period of this repeating cosmic dance, we seek out the smallest time slice where both x and y recommence their show. This search is akin to a cosmic game of 'Simon Says'—we're looking for the first time both x and y wave their hands together, signaling the full cycle's completion.