Problem 14 - College Physics

Question

At the end of her routine, an ice skater spins through 7.50 revolutions with her arms always fully outstretched at right angles to her body. If her arms are \(60.0 \mathrm{~cm}\) long, through what arc length distance do the tips of her fingers move during her finish?

Step-by-Step Solution

Answer

The arc length is approximately 28.27 meters.

Step 1: Understanding the Problem

We need to find the arc length covered by the tips of the skater's fingers during her spin. To do this, we must use the concept of arc length in a circle, which depends on the radius and the angle in radians.

Step 2: Convert Revolutions to Radians

The skater spins through 7.50 revolutions. Each revolution corresponds to an angle of \(2\pi\) radians. Therefore, the total angle in radians is calculated as follows: \[\text{Total angle} = 7.50 \times 2\pi = 15\pi \text{ radians}\]

Step 3: Identify the Radius

The radius in this problem is the length of the skater's arms, which is given as \(60.0 \text{ cm} = 0.60 \text{ m}\). This is the distance from the center of the skater's body to the tips of her fingers.

Step 4: Calculate the Arc Length

The formula for the arc length \(L\) in a circle when the angle is in radians is given by:\[L = \theta \times r\]where \(\theta\) is the angle in radians and \(r\) is the radius. Substitute the values:\[L = 15\pi \times 0.60 = 9\pi \text{ meters}\]Using the approximation \(\pi \approx 3.1416\), we find:\[L \approx 9 \times 3.1416 = 28.2744 \text{ meters}\]

Step 5: Conclusion

The arc length, which is the distance the tips of the skater's fingers move, is approximately \(28.27\) meters.

Key Concepts

Radian Measure

Radian measure offers a convenient way to express angles in mathematics and physics. Unlike degrees, a radian is based on the radius of a circle. Imagine wrapping the radius of a circle along its circumference. The angle subtended by the arc that's equal to the radius is one radian.
Radially measured angles help in simplifying calculations involving circular motion. For example, the full circle is the circumference divided by the radius, which amounts to about 6.2832 radians or simply, \(2\pi\) radians (since \(\pi \approx 3.1416\)).
This measure is dimensionless, which makes it quite handy in various applications like trigonometry and calculus.

• 1 full revolution = \(2\pi\) radians
• Conversion from revolutions to radians: multiply by \(2\pi\)
• Used often in formulas involving circular motion and periodic functions

Converting revolutions to radians is crucial for problems like the ice skater's arc length, enabling us to use simple formulas to find great lengths.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle. In physics, it's a core concept where analysis involves studying forces, velocities, and distances in a curved path.
Arc length is a key aspect of circular motion. It represents the distance traveled along the circle's edge. Often, this is where radian measure steps in to simplify the connection between an object's angular displacement and the linear distance it moves.
For the ice skater, calculating the arc length was necessary to understand how far her fingertips traveled in that circular path. Using the relationship \(\text{Arc Length} = \theta \times r\), where \(\theta\) is in radians and \(r\) is the radius, gives us a straightforward way to translate rotational motion into linear distances.

• Essential in understanding motion in circular paths
• Helps in calculating distances covered along circular arcs
• Informs dynamics like velocity and acceleration in circular paths

Interconnected with geometry and trigonometry, circular motion provides practical insights into real-world phenomena.

Geometry

Geometry, the study of shapes and sizes, plays a crucial role in problems involving circles and arcs. It helps in understanding circular arcs like those in the ice skater's motion by providing the framework in which lengths, angles, and areas are calculated.
In our problem, the geometry of the circle was used to determine the path of the skater's hand tips. With a known radius and angle, we can find how far a point travels along the curved path of a circle. The radius is simply the arm's length, and the angle was calculated in radians based on the number of spins.
This kind of problem-solving connects directly with basic geometrical principles:

• Helps identify relationships between parts of a circle
• Facilitates calculations of distances in circular designs
• Supports measurement and understanding of arcs and curves

Applications span from architecture to physics, showcasing the true power of geometric calculations in everyday scenarios.