Problem 14 - Advanced Engineering Mathematics

Question

In Problems \(7-16, \mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=\cosh t \mathbf{i}+\sinh t \mathbf{j} $$

Step-by-Step Solution

Answer

The tangential and normal components of acceleration are \( a_T = \frac{1}{2\sqrt{\cosh(2t)}} \times \sinh(2t) \) and \( a_N = \sqrt{\cosh(2t) - \left(\frac{\sinh(2t)}{2\sqrt{\cosh(2t)}}\right)^2} \).\)

Step 1: Compute the Velocity Vector

The velocity vector is the first derivative of the position vector with respect to time. Given \( \mathbf{r}(t) = \cosh t \mathbf{i} + \sinh t \mathbf{j} \), the velocity \( \mathbf{v}(t) = \frac{d}{dt}(\cosh t) \mathbf{i} + \frac{d}{dt}(\sinh t) \mathbf{j} = \sinh t \mathbf{i} + \cosh t \mathbf{j} \).

Step 2: Compute the Acceleration Vector

The acceleration vector is the derivative of the velocity vector. We have \( \mathbf{v}(t) = \sinh t \mathbf{i} + \cosh t \mathbf{j} \). Differentiating, \( \mathbf{a}(t) = \frac{d}{dt}(\sinh t) \mathbf{i} + \frac{d}{dt}(\cosh t) \mathbf{j} = \cosh t \mathbf{i} + \sinh t \mathbf{j} \).

Step 3: Compute the Magnitude of the Velocity Vector

The magnitude of the velocity vector \( \mathbf{v}(t) \) is \( |\mathbf{v}(t)| = \sqrt{(\sinh t)^2 + (\cosh t)^2} = \sqrt{\cosh(2t)} \).

Step 4: Compute Tangential Component of Acceleration

The tangential component of acceleration \( a_T \) is the derivative of the magnitude of the velocity vector: \( a_T = \frac{d}{dt}|\mathbf{v}(t)| = \frac{1}{2\sqrt{\cosh(2t)}} \times \sinh(2t) \).

Step 5: Compute Normal Component of Acceleration

The normal component of acceleration \( a_N \) can be calculated using \( a_N = \sqrt{|\mathbf{a}(t)|^2 - a_T^2} \). Here \( |\mathbf{a}(t)| = \sqrt{\cosh(2t)} \), so \( a_N = \sqrt{\cosh(2t) - \left(\frac{\sinh(2t)}{2\sqrt{\cosh(2t)}}\right)^2} \).

Key Concepts

Tangential Acceleration

In vector calculus, tangential acceleration refers to that part of the total acceleration which is responsible for changing the speed of a particle moving along a path. It is directly related to how quickly or slowly the particle speeds up or slows down. When tackling problems involving tangential acceleration, remember that it can be determined by taking the derivative of the magnitude of the velocity vector. Often, this involves using calculus to differentiate the expression concerning its components. In the original problem, the tangential acceleration \( a_T \) is obtained by differentiating the magnitude \( |abla abla(T)| \) of the velocity vector, which simplifies to \( a_T = \frac{1}{2\sqrt{\cosh(2t)}} \times \sinh(2t) \). This signifies how the speed changes as the particle moves along its path, although the direction remains unchanged.

Normal Acceleration

The normal acceleration is the component of the total acceleration that acts perpendicular to the velocity vector, thus altering the direction of the particle, rather than its speed. This results in the curved path that particles often follow. This concept is crucial in understanding the particle's trajectory within a field or space. Normal acceleration can be found using the formula:

• \( a_N = \sqrt{|abla \mathbf{a}(t)|^2 - a_T^2} \).

This formula arises from the geometry of the motion, as it separates out the directional change from the speed change. In practice, for the given problem, substituting the values of \( |abla \mathbf{a}(t)| \) and \( a_T \), we reach the expression: \( a_N = \sqrt{\cosh(2t) - \left(\frac{\sinh(2t)}{2\sqrt{\cosh(2t)}}\right)^2} \). This encapsulates how the particle's path bends.

Position Vector

The position vector \( \mathbf{r}(t) \) provides us with the particle's location in a coordinate system at any time \( t \). This vector maps out the path traced by the moving particle over time. Understanding how to work with position vectors is fundamental in analyzing any motion problem. For the exercise in question, the position vector is given by \( \mathbf{r}(t) = \cosh t \mathbf{i} + \sinh t \mathbf{j} \). This means that at any point in time \( t \), the particle's position is determined by hyperbolic functions \( \cosh t \) and \( \sinh t \). Specifically, these functions can model hyperbolic motion efficiently, portraying the phenomenon where the position evolves over time. It's crucial to differentiate the position vector to derive velocity and acceleration vectors, which lead into understanding both linear and non-linear aspects of motion.

Velocity Vector

The concept of the velocity vector is at the heart of understanding motion in vector calculus. Velocity not only tells us how fast an object is moving but also in what direction. The velocity vector is the first derivative of the position vector. For the given problem, the velocity vector \( \mathbf{v}(t) \) derived from \( \mathbf{r}(t) \) is \( \sinh t \mathbf{i} + \cosh t \mathbf{j} \). This shows how the speed changes with respect to time and direction, determining the particle’s velocity at a specific time. Proper analysis of the velocity vector helps you to determine both the magnitude and direction of motion, assisting in the computation of further derivatives such as acceleration. Understanding this can be exceptionally beneficial for problems involving dynamic systems.

Acceleration Vector

Acceleration is a measure of how quickly the velocity of a particle changes as it moves along its path. This is represented through the acceleration vector, which is the derivative of the velocity vector with respect to time. For the given exercise, the acceleration vector \( \mathbf{a}(t) \) is calculated to be \( \cosh t \mathbf{i} + \sinh t \mathbf{j} \). This represents the rate of change of the velocity vector and provides a comprehensive view of both how the speed and the direction of the motion are changing. Acceleration vectors are essential in understanding dynamics as they give insight into forces acting on the object and allow us to determine tangential and normal accelerations. Computing this vector helps to forecast the motion trajectory and behavior of the system over time effectively.