Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the tangential and normal components \(\left(a_{T}\right.\) and \(a_{N}\) ) of the acceleration vector at \(t\). Then evaluate at \(t=t_{1}\). \(\mathbf{r}(t)=a \cosh t \mathbf{i}+a \sinh t \mathbf{j} ; t_{1}=\ln 3\)

Step-by-Step Solution

Answer

At \( t = \ln 3 \), the tangential component is \( 0.8a \) and the normal component is \( 0.2a \).

Step 1: Find the Velocity Vector

The velocity vector is the first derivative of the position vector \( \mathbf{r}(t) \). Thus, \( \mathbf{v}(t) = \mathbf{r}'(t) = \frac{d}{dt}(a \cosh t) \mathbf{i} + \frac{d}{dt}(a \sinh t) \mathbf{j} = a \sinh t \mathbf{i} + a \cosh t \mathbf{j} \).

Step 2: Find the Acceleration Vector

The acceleration vector is the derivative of the velocity vector \( \mathbf{v}(t) \). So, \( \mathbf{a}(t) = \mathbf{v}'(t) = \frac{d}{dt}(a \sinh t) \mathbf{i} + \frac{d}{dt}(a \cosh t) \mathbf{j} = a \cosh t \mathbf{i} + a \sinh t \mathbf{j} \).

Step 3: Calculate the Magnitude of the Velocity

The magnitude of the velocity vector is given by \( \| \mathbf{v}(t) \| = \sqrt{(a \sinh t)^2 + (a \cosh t)^2} = a \sqrt{\sinh^2 t + \cosh^2 t} = a \cosh 2t \) as \( \cosh^2 t - \sinh^2 t = 1 \).

Step 4: Find the Tangential Component of Acceleration

The tangential component of the acceleration \( a_T \) is given by \( a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{\| \mathbf{v}(t) \|} \). Calculate \( \mathbf{v}(t) \cdot \mathbf{a}(t) = (a \sinh t)(a \cosh t) + (a \cosh t)(a \sinh t) = 2a^2 \sinh t \cosh t = a^2 \sinh 2t \). Therefore, \( a_T = \frac{a^2 \sinh 2t}{a \cosh 2t} = a \tanh 2t \).

Step 5: Find the Normal Component of Acceleration

The normal component of acceleration \( a_N \) is given by \( a_N = \sqrt{\| \mathbf{a}(t) \|^2 - a_T^2} \). First, find \( \| \mathbf{a}(t) \| = a \cosh 2t \). Thus, \( a_N = \sqrt{(a \cosh 2t)^2 - (a \tanh 2t)^2} = \sqrt{a^2 (1 - \tanh^2 2t)} = a \sech 2t \).

Step 6: Evaluate at \( t = \ln 3 \)

Substitute \( t_1 = \ln 3 \) into the expressions for \( a_T \) and \( a_N \):1. \( a_T(t_1) = a \tanh(2 \ln 3) = a \frac{9 - 1}{9 + 1} = a \frac{8}{10} = 0.8a \).2. \( a_N(t_1) = a \sech(2 \ln 3) = a \frac{2}{9 + 1} = 0.2a \).

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics focused on differentiation and integration of vector fields. In vector calculus, vectors have both magnitude and direction, represented in multiple dimensions. When dealing with physical quantities like velocity and acceleration, vectors become crucial.

Knowing how to differentiate and integrate vector functions allows us to analyze the motion of objects in space. The position vector, depending on time, is often denoted as \( \mathbf{r}(t) \). Its derivative gives us the velocity vector \( \mathbf{v}(t) \), which reveals how the position changes over time. Similarly, differentiating the velocity vector provides the acceleration vector \( \mathbf{a}(t) \), showing us how the velocity changes with time.

Understanding and manipulating these vectors is key in areas like physics and engineering, where they describe phenomena like object motion, force fields, and more. By mastering vector calculus, you embody the power to unlock a deeper comprehension of these dynamic processes.

Hyperbolic Functions

Hyperbolic functions, much like their trigonometric counterparts, are derived using exponential functions. Two common hyperbolic functions are the hyperbolic sine \( \sinh(t) \) and hyperbolic cosine \( \cosh(t) \). These functions frequently appear in physics and engineering to describe shapes like hanging cables (catenaries) and certain types of motion.

Derivatives of hyperbolic functions display patterns similar to trigonometric functions. For example, the derivative of \( \sinh(t) \) with respect to \( t \) is \( \cosh(t) \), and vice versa. Additionally, hyperbolic identities, such as \( \cosh^2 t - \sinh^2 t = 1 \), are fundamental tools. These identities aid computations, as seen when determining acceleration vectors or transforming equations.

Knowing how to apply these functions gives a better understanding of complex motion, oscillations, and various mathematical models used in scientific fields.

Tangential and Normal Acceleration

Tangential and normal components of acceleration help decompose the acceleration vector into understandable parts. These components allow us to identify how an object speeds up or changes direction at any given moment.

• The tangential acceleration \( a_T \) is associated with the change in the speed of the object along its path. It tells us how fast the object is speeding up or slowing down.
• The normal acceleration \( a_N \), also known as radial or centripetal acceleration, relates to the object's change of direction. This component points towards the center of rotation, dictating how sharply the object turns.

Formulas for these components include:

• \( a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{\| \mathbf{v}(t) \|} \), linking speeds directly along the path.
• \( a_N = \sqrt{\| \mathbf{a}(t) \|^2 - a_T^2} \), which helps isolate directional change from the overall acceleration.

By differentiating between \( a_T \) and \( a_N \), it becomes easier to analyze motion dynamics, creating more precise models of real-world situations.

Velocity and Acceleration Vectors

Velocity and acceleration vectors are central to understanding motion in vector calculus. These vectors are key to unlocking the secrets of how objects move through space.

• The velocity vector \( \mathbf{v}(t) \) is the first derivative of the position vector \( \mathbf{r}(t) \). It shows the rate of change of position, in both magnitude and direction.
• The acceleration vector \( \mathbf{a}(t) \), the derivative of the velocity vector, reveals how the velocity itself changes over time.

These vectors are instrumental in characterizing movement. By differentiating them, one gains invaluable insights into an object's journey, culled from the fundamental laws of physics. Analyzing these vectors allows one to predict future movements, helping guide everything from simple physics problems to advanced engineering designs.

With the methodological breakdown of velocity and acceleration vectors, you can apply a consistent framework to apprehend any object's motion at any point in time.