Problem 585 - Advanced Calculus

Question

What force field would account for a particle of unit mass moving around in a unit circle in the plane according to the function $$ f(t)=(\cos t, \sin t) ? $$

Step-by-Step Solution

Answer

The force field that accounts for a particle of unit mass moving in a unit circle according to the given position function \(f(t) = (\cos t, \sin t)\) is \(F(t) = (-\cos t, -\sin t)\).

Step 1: 1. Find the position function

We are given the position function of the particle as: \( f(t) = (\cos t, \sin t) \)

Step 2: 2. Find the velocity function

To find the velocity function, we need to differentiate the position function with respect to time t. Differentiating each component: \(\frac{d}{dt}(\cos t) = - \sin t \) \(\frac{d}{dt}(\sin t) = \cos t \) So, the velocity function is: \( v(t) = (- \sin t, \cos t) \)

Step 3: 3. Find the acceleration function

To find the acceleration function, differentiate the velocity function with respect to time t: \(\frac{d}{dt}(-\sin t) = -\cos t \) \(\frac{d}{dt}(\cos t) = -\sin t \) So, the acceleration function is: \( a(t) = ( -\cos t, -\sin t) \)

Step 4: 4. Apply Newton's Second Law

According to Newton's Second Law: \( F = m * a \) Since the particle has a unit mass (mass = 1), the force field will be equal to the acceleration: \( F(t) = a(t) = ( -\cos t, -\sin t) \) So, the force field that accounts for the particle moving in a unit circle according to the given position function is: \( F(t) = ( -\cos t, -\sin t) \)

Key Concepts

Particle Motion

Understanding particle motion involves tracking how a particle moves through space over time. In this exercise, the particle follows a unit circle, which is a circle with a radius of one unit. The position of the particle can be described by the function:

• \(f(t) = (\cos t, \sin t)\)

This function outlines how the particle's position changes as a function of time, \(t\).
It moves around the circumference of the circle, tracing it completely as \(t\) varies.
This type of motion is regular and predictable, reflecting the symmetrical nature of circular motion.

Differentiation

Differentiation is a fundamental calculus concept that helps us understand how functions change.
In the context of particle motion, differentiation allows us to find the velocity and acceleration functions from the position function.

• The derivative of the position function tells us how fast the position of the particle is changing, which is its velocity.
• Taking the derivative of the velocity function gives us the acceleration, which tells us how the velocity is changing over time.

In this scenario, differentiating the position function \(f(t) = (\cos t, \sin t)\) yields:

• Velocity: \(v(t) = (- \sin t, \cos t)\)
• Acceleration: \(a(t) = (-\cos t, -\sin t)\)

These derivatives help in the deeper understanding of the particle's motion around the circle.

Newton's Second Law

Newton's Second Law is key in understanding how forces affect motion.
This law states that the force exerted on an object is equal to the mass of the object times its acceleration:

• \( F = m \times a \)

In this exercise, the particle has a unit mass, meaning its mass is 1.
Applying Newton's Second Law simplifies the force exerted on the particle to being directly equal to its acceleration. Thus, we have:

• \( F(t) = a(t) = (-\cos t, -\sin t)\)

This shows that the forces acting on the particle result in its acceleration, keeping it on the circular path.

Acceleration Function

An acceleration function describes how an object's velocity changes with time, providing insight into the dynamics of motion.
In our scenario, we found the acceleration function by differentiating the velocity function. The result:

• \(a(t) = (-\cos t, -\sin t)\)

This acceleration points towards the center of the circle and is constant in magnitude, which is typical for uniform circular motion.The acceleration function helps us visualize:

• The direction of the force keeping the particle on its circular path (towards the circle's center).
• The constant rate at which the particle's velocity direction changes.

Recognizing the pattern of acceleration is vital to understanding forces in circular motion and supports the application of Newton's Second Law.