Problem 50 - Algebra 2

Question

Physics A pendulum 36 in. long swings \(30^{\circ}\) from the vertical. How high above the lowest position is the pendulum at the end of its swing? Round your answer to the nearest tenth of an inch.

Step-by-Step Solution

Answer

The pendulum is 18 inches above the lowest position at the end of its swing.

Step 1: Understanding the Pendulum Motion

Visualise the motion of the pendulum. In the question, the pendulum is swung to an angle of \(30^{\circ}\) from the vertical. Note that we are given the length of the pendulum, which is the complete distance from the pivot point to the bob (the weight at the end of a pendulum). We are asked to find the vertical height from the lowest point of swing to the point where the pendulum reaches after swinging \(30^{\circ}\) from the vertical.

Step 2: Trigonometric Concept

The right triangle is formed by the maximum swing position of the pendulum, its lowest point, and the horizontal line drawn from the maximum swing point to the vertical line. We can use the Sine function, which in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Here, the angle is \(30^{\circ}\), hypotenuse is the length of the pendulum, and the opposite side is the height we want to find.

Step 3: Applying the Trigonometric Sin Function

We plug in the values to the sin function, letting h represent the opposite side, or the vertical height above the lowest position that we are trying to find. So we have, \(\sin(30^{\circ}) = \frac{h}{36}\)

Step 4: Solving for h

Rearrange the above equation to solve for h. The resulting equation is \(h = 36 \times \sin(30^{\circ})\). The value of \(\sin(30^{\circ})\) is known as 0.5.

Step 5: Finding the height

Substitute the value of \(\sin(30^{\circ})\) in the equation \(h = 36 \times \sin(30^{\circ})\), we get \(h = 36 \times 0.5\) which equals to 18 inches.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It's a cornerstone for understanding many physical phenomena, including pendulum motion. In this particular exercise, we are dealing with a pendulum that creates a right triangle as it swings to one side. This involves an angle and provides an opportunity to apply trigonometric functions.
The essential components of trigonometry include several functions like sine, cosine, and tangent, which each have unique applications. For any given angle in a right triangle:

• The sine function (sin) relates the opposite side of the triangle to its hypotenuse.
• The cosine function (cos) relates the adjacent side to the hypotenuse.
• The tangent function (tan) accounts for the ratio of the opposite side to the adjacent side.

Understanding these relationships helps us solve problems where we know certain angles and lengths, as seen when determining the pendulum's maximum height during its swing.

Sine Function

The sine function is critical for solving problems related to pendulum motion. In the context of our problem, it gives us a way to calculate the height from the lowest point to the maximum swing position. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. This is written as:\[ ext{sin}( heta) = \frac{ ext{opposite side}}{ ext{hypotenuse}} \]In our pendulum example, the angle is given as \(30^{\circ}\), and the hypotenuse is the entire length of the pendulum, which is 36 inches. We've applied that this setup allows us to use the sine function to find the missing height, which is the side opposite the given angle.
The relationship helps us derive:\[ ext{sin}(30^{\circ}) = \frac{h}{36} \]This enables us to find the value of \( h \) by rearranging the equation, allowing us to plug in the value of \( ext{sin}(30^{\circ}) = 0.5 \). This step is essential in many practical calculations involving pendulum swings and oscillations.

Right Triangle Geometry

Right triangle geometry involves triangles where one angle is exactly \(90^{\circ}\), making them ideal for applying trigonometric principles. In our pendulum problem, we consider the segment that forms a right triangle during its swing. The right triangle consists of:

• The hypotenuse, which is the pendulum's length, at 36 inches.
• The opposite side, which is the vertical height we aim to find.
• The adjacent side, representing the horizontal distance from the pivot point to directly below the pendulum bob at rest.

In many problems, knowing just a few measurements can allow you to find the rest by using geometric principles. Right triangles are especially useful because the Pythagorean theorem and sine/cosine functions make it straightforward to solve for missing sides or angles.
For the pendulum, establishing this right triangle helped us to apply the sine function, which directly relates to the opposite side's desired height. Such triangles form the basis for many physics and engineering calculations where angular motion and static positions need to be analyzed.