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Chapter 12: Problem 43

For the helix \(\mathbf{r}=a \cos t \mathbf{i}+a \sin t \mathbf{j}+c t \mathbf{k}\), find the value of \(c(c>0)\) so that the helix will make one complete turn in a distance of 3 units measured along the \(z\) -axis.

Short Answer

Expert verified
The value of \( c \) should be \( \frac{3}{2\pi} \).

Step by step solution

01

Understand the problem statement

The helix is defined by the vector function \( \mathbf{r}(t) = a \cos t \mathbf{i} + a \sin t \mathbf{j} + c t \mathbf{k} \). We need to determine the value of \( c \) such that one complete turn of the helix corresponds to a 3-unit distance along the \( z \)-axis.
02

Determine the change in \( z \) for one complete turn

One complete turn of the helix corresponds to \( t \) changing by \( 2\pi \). For the \( z \)-component, \( z = c t \), the change in \( z \) is given by \( \Delta z = c (2\pi) - c(0) = 2\pi c \).
03

Set up the equation based on the problem's condition

According to the problem, the change in the \( z \)-component (\( \Delta z \)) for one complete turn should be 3 units. Thus, we have the equation: \( 2\pi c = 3 \).
04

Solve the equation for \( c \)

Solve the equation \( 2\pi c = 3 \) for \( c \). Dividing both sides by \( 2\pi \), we get: \( c = \frac{3}{2\pi} \).
05

Verify the solution

Verify that when \( c = \frac{3}{2\pi} \), the distance along the \( z \)-axis for one complete turn is indeed 3 units. The change in \( z \) from \( t=0 \) to \( t=2\pi \) is: \( 2\pi \left( \frac{3}{2\pi} \right) = 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Function
The concept of a vector function is central to understanding how paths like a helix are formed in three-dimensional space. A vector function, such as \( \mathbf{r}(t) = a \cos t \mathbf{i} + a \sin t \mathbf{j} + c t \mathbf{k} \), describes a vector that depends on a parameter \( t \).
This function assigns a unique vector to each value of \( t \), effectively tracing a path in space. The components of the vector function鈥擻( a \cos t \), \( a \sin t \), and \( c t \)鈥攃orrespond to the x, y, and z coordinates, respectively.
  • This means for each \( t \), the vector function describes a specific point on the helix.
  • As \( t \) varies, the entire curve of the helix is generated in the space.
This is especially useful in visualizing curves and motions that follow specific patterns.
Complete Turn
Understanding the concept of a complete turn is crucial when dealing with helical paths. In the context of a helix, a complete turn means that the path described by the vector function wraps around the central axis once.
For our helix, described by \( \mathbf{r}(t) = a \cos t \mathbf{i} + a \sin t \mathbf{j} + c t \mathbf{k} \), a complete turn corresponds to the parameter \( t \) changing by \( 2\pi \).
  • This \( 2\pi \) change allows the x and y components to complete a full circle in the xy-plane due to their trigonometric nature.
  • Consequently, the path rises following the z-component increment as it loops once around the axis.
Thus, a complete turn in a helix perfectly integrates both circular and linear motion.
Z-axis Distance
The z-axis distance in the helix equation represents the extent of vertical movement for a given change in \( t \). In the problem, we are asked to find the value of \( c \) so that one complete turn of the helix results in a specific vertical displacement along the z-axis.
The z component of the vector, \( c t \), indicates how z changes with \( t \). For one complete turn, \( t \) moves from 0 to \( 2\pi \), resulting in a z-axis change of \( \Delta z = 2\pi c \).
  • To achieve a vertical change of exactly 3 units in one complete turn, we solve for \( c \) using the equation: \( 2\pi c = 3 \).
  • This ensures that every loop around the helix advances the position by the desired z-distance.
Finding the right value of \( c \) ensures the helix behaves as expected vertically.
Parametric Equations
Parametric equations are a powerful tool in mathematics for describing curves and surfaces. For a helix, parametric equations define how each of the x, y, and z coordinates depend on a single parameter \( t \).
In our helix, the parametric equations are \( x(t) = a \cos t \), \( y(t) = a \sin t \), and \( z(t) = c t \).
  • Each equation determines how a specific coordinate changes as \( t \) varies.
  • The x and y equations involve sine and cosine, which create the circular path in the xy-plane.
  • The z equation adds the vertical dimension by linearly increasing with \( t \).
This combination of trigonometric and linear functions allows for the description of more complex three-dimensional paths like a helix.

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Most popular questions from this chapter

Find the velocity, speed, and acceleration at the given time \(t\) of a particle moving along the given curve. I" $$ \mathbf{r}(t)=t \mathbf{i}+\frac{1}{2} t^{2} \mathbf{j}+\frac{1}{3} t^{3} \mathbf{k} ; t=1 $$

Determine whether the statement is true or false. Explain your answer. If the line \(y=x\) is parametrized by the vector-valued function \(\mathbf{r}(t)\), then \(\mathbf{r}(t)\) is smooth.

Assume that \(s\) is an arc length parameter for a smooth vector-valued function \(\mathbf{r}(s)\) in 3 -space and that \(d \mathbf{T} / d s\) and \(d \mathbf{N} / d s\) exist at each point on the curve. (This implies that \(d \mathbf{B} / d s\) exists as well, since \(\mathbf{B}=\mathbf{T} \times \mathbf{N}\).) The following derivatives, known as the Frenet-Serret formulas, are fundamental in the theory of curves in 3 -space: $$ \begin{aligned} d \mathbf{T} / d s &=\kappa \mathbf{N} & &[\text { Exercise } 58] \\ d \mathbf{N} / d s &=-\kappa \mathbf{T}+\tau \mathbf{B} & &[\text { Exercise } 60] \\ d \mathbf{B} / d s &=-\tau \mathbf{N} & &[\text { Exercise 59(c) }] \end{aligned} $$ Use the first two Frenet-Serret formulas and the fact that \(\mathbf{r}^{\prime}(s)=\mathbf{T}\) if \(\mathbf{r}=\mathbf{r}(s)\) to show that \(\tau=\frac{\left[\mathbf{r}^{\prime}(s) \times \mathbf{r}^{\prime \prime}(s)\right] \cdot \mathbf{r}^{\prime \prime \prime}(s)}{\left\|\mathbf{r}^{\prime \prime}(s)\right\|^{2}} \quad\) and \(\quad \mathbf{B}=\frac{\mathbf{r}^{\prime}(s) \times \mathbf{r}^{\prime \prime}(s)}{\left\|\mathbf{r}^{\prime \prime}(s)\right\|}\)

In these exercises \(\mathbf{r}(t)\) is the position vector of a particle moving in the plane. Find the velocity, acceleration, and speed at an arbitrary time \(t\). Then sketch the path of the particle together with the velocity and acceleration vectors at the indicated time \(t\) $$ \mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j} ; t=0 $$

Determine whether the statement is true or false. Explain your answer. If the smooth vector-valued function \(\mathbf{r}(s)\) is parametrized by arc length and \(\mathbf{r}^{\prime \prime}(s)\) is defined, then \(\mathbf{r}^{\prime}(s)\) and \(\mathbf{r}^{\prime \prime}(s)\) are orthogonal vectors.
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