Question

A revolving light sends out a bundle of rays that are approximately parallel, its distance from the shore, which is a straight beach, being half a mile, and it makes one revolution in a minute. Find how fast the light is travelling along the beach when at the distance of a quarter of a mile from the nearest point of the beach.

Short answer

Answer: The speed of the light along the beach when it is a quarter-mile from the closest point is \(\frac{\sqrt{3}\pi}{2} \, \text{miles per minute}\).

Step-by-step solution

Draw a diagram and define variables

Begin by drawing a diagram to represent the scenario, including the light, the beach, and the distances given. Let the center of the rotation be O, the point where the light is closest to the beach be B, and the point where the light is at when it's a quarter-mile from the beach be A. Define the following variables: \(OB = \frac{1}{2}\, \text{mile}\) (distance of the light from the shore) \(AB = \frac{1}{4}\, \text{mile}\) (distance of the light from the closest point on the beach) \(t = 1\, \text{min}\) (time for one complete revolution)

Find the angle \(\theta\)

Determine the angle \(\theta\) formed by the triangle OAB. Note that triangle OAB is a right triangle due to the parallel rays, with OB as the hypotenuse and AB as one of the legs. Using the Pythagorean theorem, we can find the other leg OA, and then determine \(\theta\) using trigonometry: \(OA = \sqrt{OB^2 - AB^2} = \sqrt{\left(\frac{1}{2}\right)^2 - \left(\frac{1}{4}\right)^2} = \frac{\sqrt{3}}{4}\, \text{mile}\) Now that we have OA, we can use the sine function to find \(\theta\): \(\sin \theta = \frac{AB}{OB} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}\) \(\theta = \sin^{-1}\left(\frac{1}{2}\right) = 30^{\circ}\)

Calculate angular speed

Calculate the angular speed \(\omega\) defined by the relation between the angle covered in a revolution and the time spent. Since the light completes one revolution in a minute (t=1 min), its angular speed is: \(\omega = \frac{360^{\circ}}{1\, \text{min}} = 360^{\circ}\, \text{per min}\)

Find the linear speed

Calculate the linear speed \(v\) of the light along the beach when it is a quarter-mile from the closest point (point A) using the formula: \(v = \omega \cdot OA\) First, let's convert the angular speed to radians per minute: \(\omega = 360^{\circ}\, \text{per min} \times \frac{\pi \, \text{radians}}{180^{\circ}} = 2\pi\, \text{radians per min}\) Now, compute the linear speed: \(v = (2\pi\, \text{radians per min}) \cdot \left(\frac{\sqrt{3}}{4}\, \text{mile}\right) = \frac{\sqrt{3}\pi}{2} \, \text{miles per minute}\) So, the speed of the light along the beach when at the distance of a quarter of a mile from the nearest point of the beach is \(\frac{\sqrt{3}\pi}{2} \, \text{miles per minute}\).

Key concepts

Angular Speed

Angular speed is a measure of how quickly an object rotates around a point, and it's a fundamental concept in problems involving circular motion, like our revolving light problem.
It's defined as the angle through which an object rotates in a certain period of time. In mathematical terms, it's expressed as the angle divided by the time period over which the rotation occurs.

• For instance, in our exercise, the light completes one full rotation, or 360°, in one minute, giving it an angular speed of 360° per minute.
• We can also express angular speed in radians, another angle measurement where a full circle is 2π radians, so 360° converts to 2π radians.

Making these conversions is essential because radians are crucial when using formulas in calculus and physics. Hence, understanding and calculating angular speed is the first step towards solving how the light travels along the beach.

Trigonometry

Trigonometry is the branch of mathematics dealing with the relationships between the sides and angles of triangles.
It's particularly useful in this problem as the movement of the light forms a right triangle when projected onto the beach.

• To solve for variable relationships, the Pythagorean theorem helps us find unknown side lengths in a right triangle, as we see with the side OA.
• Before trigonometric functions, like the sine function, help determine the angles and further calculations, like finding the angle \(\theta\) from the known side lengths OB and AB.

In our example, by calculating the sine of the angle \(\theta\), we determined it to be 30°—crucial information to not only validate OA's length in the triangle, bringing precision and ease to subsequent steps in our solution. Grasping basic trigonometry concepts thus simplifies understanding complex problems involving geometry and calculus.

Linear Speed Calculation

Linear speed refers to how fast an object travels along a path. To find linear speed from angular speed, you need to apply some key concepts of circular motion.
In our problem, we're interested in the speed of a revolving light as it creates a path on the beach.

• The formula \(v = \omega \times r\) relates linear (\(v\), and here \(\omega\) is the previously calculated angular speed, whereas \(r\) (radius) is the perpendicular distance to the point of interest—in this case, OA.
• Since \(\omega\) is converted to radians (crucial for these calculations), multiplying by the radius gives the linear speed \(v\) in miles per minute.

Understanding this formula helps connect angular motion with straight-line speed, a conversion important for describing motion in real-world contexts. Here, it concludes with the light's speed being \(\frac{\sqrt{3}\pi}{2} \) miles per minute when it's at the specified quarter-mile point from the beach.