Problem 74 - Algebra and Trigonometry with Analytic Geometry

Question

Trigonometric functions are used extensively in the design of industrial robots. Suppose that a robot's shoulder joint is motorized so that the angle \(\theta\) increases at a constant rate of \(\pi / 12\) radian per second from an initial angle of \(\theta=0\). Assume that the elbow joint is always kept straight and that the arm has a constant length of 153 centimeters, as shown in the figure. (a) Assume that \(h=50 \mathrm{~cm}\) when \(\theta=0\). Construct a table that lists the angle \(\theta\) and the height \(h\) of the robotic hand every second while \(0 \leq \theta \leq \pi / 2\). (b) Determine whether or not a constant increase in the angle \(\theta\) produces a constant increase in the height of the hand. (c) Find the total distance that the hand moves.

Step-by-Step Solution

Answer

(a) Table shows \( h \) increases non-linearly. (b) \( h \) doesn't increase constantly. (c) Total distance is \( 76.5\pi \) cm.

Step 1: Understand the Problem

We need to create a table that shows the change in the angle \( \theta \) and the corresponding height \( h \) of the robotic hand from time \( t = 0 \) to \( t \) when \( \theta = \pi/2 \). \( \theta \) increases at a constant rate of \( \pi/12 \) radians per second, and the arm length is 153 cm.

Step 2: Calculate Time for \( \theta = \pi/2 \)

\( \theta = \frac{\pi}{12} t \) where \( \theta \) is the angle in radians and \( t \) is time in seconds. We need to find the time it takes for \( \theta \) to reach \( \pi/2 \):\[ t = \frac{\pi/2}{\pi/12} = 6 \text{ seconds} \]This shows the table will have 7 entries (from 0 to 6 seconds).

Step 3: Construct the Table for \( \theta \) and \( h \)

Since \( h \) depends on the sine of the angle because the arm is rotating upward, \( h = 153 \sin(\theta) + 50 \). We calculate \( h \) at each second:- \( t = 0 \text{ s}, \theta = 0, h = 50 \) cm.- \( t = 1 \text{ s}, \theta = \frac{\pi}{12}, h = 153 \sin\left(\frac{\pi}{12}\right) + 50 \).- \( t = 2 \text{ s}, \theta = \frac{\pi}{6}, h = 153 \sin\left(\frac{\pi}{6}\right) + 50 \).- Continue this until \( t = 6 \text{ s}, \theta = \frac{\pi}{2} \).

Step 4: Constant Increase Evaluation

Check if the increase in \( h \) is constant. Since \( \sin(\theta) \) is non-linear, \( h = 153 \sin(\theta) + 50 \) increases non-linearly even though \( \theta \) increases at a constant rate.

Step 5: Calculate Total Distance Moved

The path described by \( h \) is a part of a circle's arc with radius 153 cm. The total angular displacement \( \Delta \theta = \frac{\pi}{2} \) radians. The arc length is given by \( 153 \times \Delta \theta = 153 \times \frac{\pi}{2} = 76.5\pi \text{ cm} \).

Key Concepts

Industrial Robots

Industrial robots are revolutionizing the manufacturing and automation industries. They are complex systems designed to perform a variety of tasks with precision and efficiency. One crucial component of these robots is their joints, such as shoulder and elbow joints, which enable movement and flexibility.
The shoulder joint in our problem is motorized to increase the angle at a constant rate, allowing the robotic arm to move smoothly and predictably. This kind of control is fundamental in applications where consistency and accuracy are key, such as in assembling products or handling delicate items.
By understanding the mechanics of industrial robots, particularly how joint angles can be manipulated, we can appreciate the role of trigonometric functions in optimizing robotic performance. They help engineers design robots that operate effectively in varied environments, improving productivity and precision in industrial settings.

Angle Measurement

Angles play a significant role in robotics, specifically in how robots are programmed to move. Understanding angles and their measurement is vital to controlling robotic joints. In our example, the angle measurement is in radians, which is a standard in engineering for its simplicity and naturalness in mathematical formulas.
By manipulating the angle \(\theta\), which increases at a constant rate, engineers can predict the trajectory of a robot's arm. An accurate measure of the angle ensures the robot can perform its task without error.
For this exercise, calculating angles in increments of \( \pi/12\) radians per second determines the new position of the robotic hand every second. This precise control allows for smooth movement, which is crucial in tasks that require exact placement or movement of objects.

Sinusoidal Functions

Sinusoidal functions, such as sine and cosine, describe the oscillatory nature of various phenomena, including robot movements. In our problem, the height \( h \) of the robotic hand is expressed as a function of sine: \( h = 153 \sin(\theta) + 50 \).
This relationship highlights how the vertical position of the robotic hand varies with the angle. The sinusoidal function captures the upward and downward movement as the arm rotates around the shoulder joint.
Sine functions are especially valuable in robotics for modeling cyclic movements, enabling smooth transitions without sudden stops or starts. These functions ensure movements are fluid, thus maintaining the efficiency and harmony required in a manufacturing line.

Radians and Degrees

Radians and degrees are two units used to measure angles, with radians being the preferred unit in mathematics and engineering due to their direct relationship with arc length.
One important aspect to understand is how converting between radians and degrees works. A full circle is \(2\pi\) radians, or 360 degrees, which makes \( \pi/2 \) radians equivalent to 90 degrees. \( \pi/12 \) radians represents 15 degrees, providing a clear framework for measurement conversion.
In our scenario, using radians simplifies the calculation of arc lengths and angle increments, providing a straightforward approach to determine how the robot moves through space. By understanding the conversion and application of radians, students can better grasp the mechanics of rotational systems and the mathematical elegance that defines them.