Question

Numerical and Graphical Reasoning A crossed belt connects a 20 -centimeter pulley \((10-\mathrm{cm} \text { radius) on an electric }\) motor with a 40 -centimeter pulley \((20-\mathrm{cm} \text { radius) on a saw }\) arbor (see figure). The electric motor runs at 1700 revolutions per minute. (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor? (c) Let \(L\) be the total length of the belt. Write \(L\) as a function of \(\phi,\) where \(\phi\) is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley.) (d) Use a graphing utility to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline \phi & {0.3} & {0.6} & {0.9} & {1.2} & {1.5} \\ \hline L & {} & {} & {} & {} \\ \hline\end{array} $$ (e) Use a graphing utility to graph the function over the appropriate domain. (f) Find \(\lim _{\phi \rightarrow(\pi / 2)^{-}} L .\) Use a geometric argument as the basis of a second method of finding this limit. (g) Find \(\lim _{\phi \rightarrow 0^{+}} L\)

Short answer

The revolution speed of the saw is 850 revolutions per minute. A crossed belt causes the saw to rotate in the opposite direction relative to the motor. For the function \(L(\phi)\), the domain is (0, π/2). The values of L for the corresponding φ values can be computed using a graphing tool. Similarly, the graph can be plotted accordingly. The limit of the function L as φ approaches \((\pi / 2)^{-}\) corresponds to twice the perimeter of rectangle formed by the edges of the pulleys. Whereas as φ approaches 0 from the left, the belt length equals twice the sum of the pulleys' circumferences.

Step-by-step solution

Calculate the saw's revolutions per minute

Notice that the pulley diameters are inversely proportionate to their rotation speeds. Let's denote the motor's speed as \(M\) and the saw's speed as \(S\). We can set up the ratio as follows: \[ \frac{M}{S} = \frac{d_{S}}{d_{M}} \] Where \(d_{M}\) and \(d_{S}\) are the diameters of the motor's and saw's pulleys respectively. Substituting the given values, we get: \[ \frac{1700}{S} = \frac{20}{10} \] Solving for \(S\), we obtain the speed of the saw.

Evaluate effect of crossed belt on saw operation

The belt is crossed to change the direction of the saw. This means that the saw rotates in the opposite direction as the motor rotates.

Define L as a function of φ

The length of the belt L can be expressed as: \[L = 2 \sqrt{20^2 - 10^2 \cos \phi} + 10 \phi + 20 (\pi - \phi)\] This equation is derived from the geometry of the setup, taking into account the straight sections of the belt and the belt length around each pulley. As per the given problem, the valid values of \(φ\) lie between 0 (exclusive) and \(\pi / 2\) (exclusive), forming the domain of the function.

Use the graphing utility to complete the table

Instructions on how to use the graphing utility will vary, but typically involve entering the defined function \(L\) and using the given table values for \(φ\). Use the function from Step 3 to replace \(L\).

Graph the function

Enter the equation from Step 3 into the graphing utility and specify the domain to get the graph.

Find the limit of the function as φ approaches \((\pi / 2)^{-}\)

For \(\lim _{\phi \rightarrow(\pi / 2)^{-}} L\), a geometric reasoning implies that as \(φ\) approaches from left to \((\pi / 2)^{-}\), the belt straightens out and frames a rectangle, and the length of the belt is twice the perimeter of the rectangle formed by the edges of the pulleys.

Find the limit of the function as φ approaches \(0^{+}\)

For \(\lim _{\phi \rightarrow 0^{+}} L\), as \(φ\) approaches from right to 0, the straight sections of the belt coincide with the line joining the centers of the pulleys. The belt length becomes 2 times sum of the circumferences of the pulleys.

Key concepts

Numerical and Graphical Reasoning

Numerical and graphical reasoning are pivotal in understanding complex concepts in calculus and other scientific fields. In the context of pulleys and rotational mechanics, numerical reasoning helps us quantify the relationships between variables such as pulley radius, belt length, and rotation speed. For instance, when we know the speed of the motor and the size of its pulley, we can use numerical methods to calculate the speed at which a connected saw will rotate.

Graphical reasoning, on the other hand, allows us to visualize these relationships. A graph can show how varying the angle \( \phi \) influences the total length of the belt in our textbook problem. By plotting these functions, we can observe trends, notice behavior near certain values, and help predict the behavior of our pulley system under different conditions. A more nuanced understanding is gained by linking the tangible visual graph to the abstract numerical data.

Function of Phi (φ)

Understanding Phi in Rotational Mechanics

In rotational mechanics, the angle \( \phi \) is often used to represent the angular position of a point, or in our case, the angular component of the belt around a pulley. Our textbook exercise asks us to express the total length \(L\) of a crossed belt as a function of \( \phi \).

This function incorporates the geometry of the pulley system and includes components for the straight sections of the belt as well as the parts of the belt that wrap around the pulleys. By understanding how \( \phi \) is used in this context, students can better grasp how changes in one part of a system can affect other parts, leading to a more comprehensive understanding of systems in motion.

RPM and Pulley Diameter Relationship

Linking RPM and Pulley Size

The revolutions per minute (RPM) of a pulley is inversely related to its diameter - a concept that can be counterintuitive at first glance. In our exercise, we have two pulleys of different diameters connected by a belt, and this relationship tells us that the larger pulley will rotate slower than the smaller one.

To calculate the RPM of the saw, we start with the known RPM of the motor and apply the principle that the product of the RPM and the pulley diameter is constant along the belt. This principle is rooted in the conservation of angular velocity and ensures the continuous motion of the belt. In educational content, making these connections clear and relating them to conservation laws helps demystify why such a relationship exists.

Graphing Utility Applications

Real-World Uses of Graphing Utilities

Graphing utilities are not just abstract educational tools; they have real-world applications in engineering, economics, and many other fields. Applying a graphing utility to our problem, students learn to manipulate variables, interpret results, and predict outcomes.

In our exercise, students are instructed to use a graphing utility to populate a table of values and to graph the function of \(L\) in relation to \( \phi \). This hands-on experience with technology not only aids in comprehension but also prepares students for future scenarios where they may need to use similar tools to solve real-world problems. It's essential for educational material to highlight the practicality of such tools beyond the classroom.

Limit of a Function

Conceptualizing Limits in Mechanical Systems

The concept of a limit in calculus is about understanding behavior as one quantity gets arbitrarily close to a specific value. In the case of our pulley system, we are interested in the limit of \(L\) as \( \phi \) approaches either \( \frac{\pi}{2} \) (from the left) or \(0\) (from the right).

The limit helps us determine the maximum length the belt can reach given the constraints of the pulleys, or what happens to the length when the angle \( \phi \) becomes very small. In this context, students can witness how abstract mathematical concepts such as limits directly apply to physical systems and learn to reconcile geometric logic with algebraic calculation to arrive at the same conclusion, enriching their problem-solving toolkit.