If a carnival swing is fixed on the end of an 8.1-m-long beam. The swing and beam sweep through an angle of 120 degrees. How do i find the distance through which the riders move???

To find the distance that the riders move on a carnival swing fixed at the end of an 8.1-meter beam, we can visualize the swing's path as part of a circular arc. The key here is to determine the length of that arc, which is defined by the angle through which the swing moves. Let's break this down step by step.

Understanding the Components

The swing is attached to a beam that acts as a radius of a circle. In this case, the radius (r) is the length of the beam, which is 8.1 meters. The swing sweeps through an angle of 120 degrees. To find the distance traveled by the riders, we need to calculate the arc length of the circle corresponding to that angle.

Formula for Arc Length

The formula for the arc length (L) of a circle is given by:

• L = r × θ

Where:

• L is the arc length.
• r is the radius of the circle.
• θ is the angle in radians.

Converting Degrees to Radians

Since the angle is provided in degrees, we need to convert it to radians. The conversion factor is:

• 1 radian = 180/π degrees

To convert 120 degrees to radians:

• θ (in radians) = 120 × (π/180) = 2π/3 radians

Calculating the Arc Length

Now that we have the radius and the angle in radians, we can substitute these values into the arc length formula:

• L = 8.1 m × (2π/3)

Calculating this gives:

• L ≈ 8.1 m × 2.0944 ≈ 16.95 m

Final Result

The distance through which the riders move on the carnival swing is approximately 16.95 meters. This distance represents the length of the arc that the swing covers as it moves through the 120-degree angle.

Visualizing the Motion

To better understand this, imagine standing at the center of a circle with a radius of 8.1 meters. As the swing moves from one side to the other, it traces out a curved path. The longer the radius or the larger the angle, the greater the distance traveled along that arc. This concept is fundamental in circular motion and helps us appreciate how swings and rides operate in amusement parks.