Problem 2 - College Physics

Question

\(\bullet\) (a) What angle in radians is subtended by an arc 1.50 \(\mathrm{m}\) in length on the circumference of a circle of radius 2.50 \(\mathrm{m} ?\) What is this angle in degrees? (b) An arc 14.0 \(\mathrm{cm}\) in length on the circumference of a circle subtends an angle of \(128^{\circ} .\) What is the radius of the circle? (c) The angle between two radii of a circle with radius 1.50 \(\mathrm{m}\) is 0.700 rad. What length of arc is intercepted on the circumference of the circle by the two radii?

Step-by-Step Solution

Answer

(a) 0.60 radians; 34.38°. (b) Radius is 6.27 cm. (c) Arc length is 1.05 m.

Step 1: Find the angle in radians for part (a)

Given the length of the arc \(s = 1.50\) m and the radius \(r = 2.50\) m. The angle \(\theta\) in radians that is subtended by this arc can be found using the formula \(s = r\theta\). Rearranging for \(\theta\), we get \(\theta = \frac{s}{r} = \frac{1.50}{2.50} = 0.60\) radians.

Step 2: Convert radians to degrees for part (a)

To convert an angle in radians to degrees, use the conversion factor \(\frac{180}{\pi}\). The angle in degrees is \(\theta_{\text{degrees}} = 0.60 \times \frac{180}{\pi} \approx 34.38^{\circ}\).

Step 3: Find the circle radius for part (b)

Given the length of the arc \(s = 14.0\) cm and the angle \(\theta = 128^{\circ}\). First, convert the angle from degrees to radians using \(\theta_{\text{radians}} = 128 \times \frac{\pi}{180}\). This gives \(\theta_{\text{radians}} \approx 2.234\) radians. Now, use the formula \(s = r\theta\) to solve for the radius \(r\). Rearrange to get \(r = \frac{s}{\theta} = \frac{14.0}{2.234} \approx 6.27\) cm.

Step 4: Find the arc length for part (c)

The angle between two radii is \(\theta = 0.700\) radians with a radius of \(r = 1.50\) m. Use the formula \(s = r\theta\) to find the arc length: \(s = 1.50 \times 0.700 = 1.05\) m.

Key Concepts

Arc Length

The arc length is the distance measured along the curved line forming part of the circle's circumference. It helps in understanding how much of the circle's perimeter is covered by the arc in question.

To calculate the arc length, you need to know the circle's radius and the angle subtended by the arc at the circle's center. The formula used is \[ s = r\theta \]where:

• \(s\) represents the arc length,
• \(r\) is the radius of the circle, and
• \(\theta\) denotes the angle in radians.

For example, if you have an arc that is part of a circle with a radius of 1.50 meters and the angle subtended is 0.70 radians, the arc length \(s\) would be given by multiplying the radius and the angle: \[s = 1.50 \times 0.70 = 1.05 \text{ meters}\].

Understanding arc length is a vital part of learning about circle geometry as it connects the concept of angles to the real measurement of curves in the circle.

Circle Radius

The radius of a circle is the distance from its center to any point on its circumference. The radius plays a key role in various calculations related to circles, including finding the area, circumference, and arc length.

To find the radius when you know the arc length and the angle the arc subtends at the circle's center, you can rearrange the arc length formula. Starting from \[ s = r\theta \]you can solve for the radius \(r\):\[ r = \frac{s}{\theta} \]For instance, if the arc length is 14.0 cm and the subtended angle is 2.234 radians (after conversion from degrees), the radius can be determined as: \[r = \frac{14.0}{2.234} \approx 6.27 \text{ cm}\].

Having the radius is crucial because it allows you to compute other important circle properties and understand the dimensions of the circle layout.

Radians to Degrees Conversion

Radians and degrees are two units for measuring angles. While degrees are more commonly used in day-to-day applications, radians offer a natural way to express angles in mathematical calculations, especially in the context of circle geometry.

To convert an angle from radians to degrees, you use the conversion factor \[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]This formula converts the measure of the angle from the circle's arc to its equivalent in degrees. For instance, if you have an angle of 0.60 radians, its degree equivalent is calculated as: \[0.60 \times \frac{180}{\pi} \approx 34.38^{\circ}\].

Understanding this conversion is essential for translating problems into a more familiar context, which can simplify solving circle-related exercises and understanding the geometric concepts involved.

Circle Geometry

Circle geometry delves into the many properties and measurements related to circles. It encompasses concepts such as radius, diameter, circumference, angles, and the relationships between them.

Some important points in circle geometry include:

• The circumference \(C\) of a circle is given by \(C = 2\pi r\), where \(r\) is the radius.
• An angle measured at the circle's center is made by two radii, creating an arc on the circle's perimeter.
• Arcs, sectors, and segments are various parts of a circle that hold significance for different practical applications.

Understanding these relationships and properties is fundamental when solving problems involving circles. Circle geometry isn't just confined to theoretical math; it applies to real-world scenarios such as engineering, navigation, and design, where precise angle and length calculations are imperative for accuracy and functionality.