Problem 31 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Use a calculator to solve if necessary. The angular acceleration \(\alpha\) (in \(\mathrm{rad} / \mathrm{s}^{2}\) ) of the wheel of a car is given by \(\alpha=-0.2 t^{3}+t^{2},\) where \(t\) is the time (in s). For what values of \(t\) is \(\alpha=2.0 \mathrm{rad} / \mathrm{s}^{2} ?\)

Step-by-Step Solution

Answer

The values of \( t \) at which \( \alpha = 2.0 \mathrm{rad} / \mathrm{s}^{2} \) are found by solving the cubic equation \( -0.2t^3 + t^2 - 2.0 = 0 \).

Step 1: Identify the problem equation

We start with the given equation for angular acceleration: \( \alpha = -0.2t^3 + t^2 \). We need to find the value of \( t \) when \( \alpha = 2.0 \mathrm{rad} / \mathrm{s}^{2} \).

Step 2: Set up the equation

We set the equation for \( \alpha \) equal to 2.0: \( 2.0 = -0.2t^3 + t^2 \). This will allow us to solve for \( t \).

Step 3: Rearrange the equation

Rearrange the equation to form a polynomial: \( 0 = -0.2t^3 + t^2 - 2.0 \). Now, it is in a standard polynomial form \( 0 = at^3 + bt^2 + c \).

Step 4: Solve the polynomial equation

Use a calculator or a computational method to find the roots of the cubic equation. The cubic equation is \( -0.2t^3 + t^2 - 2.0 = 0 \).

Step 5: Identify real solutions

When solving the equation \( -0.2t^3 + t^2 - 2.0 = 0 \), you find the real solutions. These are the times at which the angular acceleration is \( 2.0\mathrm{rad} / \mathrm{s}^{2} \).

Key Concepts

Polynomial Equations

A polynomial equation is an expression of the form \[a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0\], where the terms are composed of variables (\( x \)) raised to integer powers and multiplied by coefficients \( a_n \). In our problem, the polynomial equation is \(-0.2t^3 + t^2 - 2.0 = 0\). Each term represents a part of the equation involving the variable \( t \), which is raised to the power corresponding to its degree.
The degree of a polynomial is determined by the highest power of the variable present in the equation. In this case, the degree is 3, meaning it's a cubic polynomial equation. Cubic polynomials can have up to three real roots. The challenge is to find the values of \( t \) that satisfy or "solve" the equation, essentially yielding values where the equation equals 0.

Angular Acceleration

Angular acceleration describes how quickly an object's rotational speed changes. It is denoted as \(\alpha\) and is typically measured in radians per second squared \((\text{rad/s}^2)\).
In the given problem, the angular acceleration is specified by the polynomial function \(\alpha=-0.2t^3+t^2\). This tells us how \(\alpha\) changes over time \(\text{t}\), impacting the rotational motion of the car's wheel. When the problem asks us to find when \(\alpha = 2.0\text{ rad/s}^2\), we're looking for the specific instants in time when the wheel's angular acceleration is exactly \(2.0 \text{ rad/s}^2\). Understanding this concept is crucial for analyzing systems involving rotational dynamics, like engines and wheels, where precise timing and acceleration can significantly impact performance.

Problem Solving

Problem solving in calculus often involves setting up and manipulating equations to find unknown values of variables. The process begins by clearly understanding the given equation that represents the physical situation—in this case, the angular acceleration equation \(-0.2t^3 + t^2 = 2\).
To successfully solve for \(t\), we follow a series of logical steps:

• Identify the given equation or function that needs to be solved.
• Set the function equal to the desired value, \(2.0\text{ rad/s}^2\), to form the equation needing a solution.
• Reorganize the equation into a standard numerical form, which aids in solving using algebraic or computational techniques.
• Find the roots (solutions) of the equation to determine when the desired state occurs.

By understanding each step's purpose and function, problem solving becomes a structured method for approaching complex mathematical scenarios and applications.

Real Solutions

Real solutions refer to the actual numerical values that solve an equation where no imaginary numbers are involved. In the polynomial \(-0.2t^3 + t^2 - 2.0 = 0\), we're searching for the real values of \(t\) where the equation holds true.
These real solutions represent times when the car's wheel has an angular acceleration of \(2.0 \text{ rad/s}^2\). It’s important in physical problems since real solutions correspond to practical, measurable events or situations. For polynomials, especially those of degree 3, solving them can lead to multiple solutions, not all of which may be real. The task is to extract the realistic values that apply to the situation being modeled. This often involves graphing or using numerical methods to approximate solutions when algebraic methods are cumbersome or not possible.