Problem 58 - Calculus Early Transcendentals: Pearson New International Edition

Question

An object moves along the curve \(y=\sin 2 x\). Without doing any calculating, decide where \(a_{N}=0\).

Step-by-Step Solution

Answer

The normal acceleration \(a_N = 0\) at inflection points (e.g., \(x = 0, \pi, 2\pi, \ldots\)).

Step 1: Understanding the Problem

We need to determine where the normal acceleration, denoted as \(a_N\), is zero for an object moving along the curve \(y=\sin 2x\). The normal acceleration is related to the curvature of the path. Where the path is straight, \(a_N\) becomes zero.

Step 2: Identifying Zero Curvature Points

A curve \(y=\sin 2x\) will have a straight path when the slope of the tangent line is constant because the curvature depends on the rate of change of the slope. Since \(\sin 2x\) has its slope zero at the turning points (like 0, \(\pi\), \(2\pi\), etc.), these points create inflection points where curvature can be zero.

Step 3: Confirming Zero Curvature

At these turning points, the change in slope is neither increasing nor decreasing, meaning the path behaves momentarily like a straight line, making the normal acceleration zero. Thus, these are the points where \(a_N = 0\).

Key Concepts

Normal Acceleration

Normal acceleration (\(a_N\)) is a concept in physics and calculus that reflects how an object changes direction when moving along a curved path. It is one of two components of acceleration experienced by an object in motion, with the other being tangential acceleration. While tangential acceleration concerns changes in speed along the path, normal acceleration is all about the change in direction.

To imagine this, think about the motion of a car going around a curve. Even if the speedometer doesn't change, the car feels a force pushing it sideways because the direction is constantly changing. This is normal acceleration at work.

• Normal acceleration becomes zero when the path of motion is straight.
• In calculus, this occurs at points where the curvature of the path is zero.
• For the equation \(y = \sin 2x\), we identify these points by looking at the nature of the curve.

Whenever you see a straight part of a path, such as certain positions within a wave-like sinusoidal curve, expect normal acceleration to drop to zero.

Inflection Points

Inflection points on a curve are special places where the curve changes its direction in terms of concavity. It is where the curve switches from curving upwards (concave up) to curving downwards (concave down) or vice versa. At these points, the curvature of the line changes, often causing the tangent line to swap its slope direction.

In the context of the trigonometric function \(y = \sin 2x\), inflection points are crucial because:

• These points correspond to places where the curvature can be zero.
• The slope of the tangent line stops increasing or decreasing momentarily.
• This corresponds with the points where normal acceleration, \(a_N\), becomes zero.

To understand inflection points visually, think of a rollercoaster cresting over a hill: when it goes from climbing to descending. That's an inflection point in terms of motion on the track of a curve.

Trigonometric Functions

Trigonometric functions like \(\sin, \cos,\) and \(\tan\) are essential in studying periodic patterns and oscillations. These functions are wave-like and repetitive, offering unique insights into the behavior of positions and velocities at different points on the wave.

For the function \(y = \sin 2x\), understand:

• It oscillates between \(-1\) and \(1\) and completes a cycle with every increment of \(\pi\).
• The slope of the curve changes continuously, except at specific points.
• These points are crucial for understanding concepts like normal acceleration and where it becomes zero.

Recognizing the properties of sine and other trigonometric functions allows mathematicians to model and predict physical phenomena, such as harmonics or oscillatory motion represented by \(\sin 2x\).

Slope of Tangent Line

The slope of the tangent line to a curve at any given point tells us about the instantaneous rate of change at that point. In the calculus of motion, it's the first derivative of the curve function, representing how steep or flat it is at any point.

Considering the function \(y = \sin 2x\):

• The slope is found by calculating the derivative, which is \(2 \cos 2x\).
• Points where the derivative is zero (i.e., where \(2 \cos 2x = 0\)) indicate horizontal tangent lines.
• These are important as they relate to inflection points and suggest where normal acceleration might be zero.

Therefore, understanding the slope of tangent lines helps us determine how the direction of a moving object changes and leads to insights about the object's speed and when it may have zero curvature or normal acceleration.