Problem 2 - College Physics

Question

(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 are is intercepted on the circumference of the circle by the two radii?

Step-by-Step Solution

Answer

(a) 0.60 radians, 34.38°; (b) 6.27 cm; (c) 1.05 m.

Step 1: Convert Arc Length to Radians

To find the angle in radians subtended by an arc, use the formula \( \theta = \frac{s}{r} \), where \( s \) is the arc length and \( r \) is the radius. Here, \( s = 1.50 \, \text{m} \) and \( r = 2.50 \, \text{m} \). Calculate \( \theta \):\[\theta = \frac{1.50}{2.50} = 0.60 \, \text{radians}\]

Step 2: Convert Radians to Degrees

To convert the angle from radians to degrees, use the conversion \( 1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees} \). Thus:\[\theta = 0.60 \, \text{radians} \times \frac{180}{\pi} \approx 34.38^{\circ}\]

Step 3: Calculate Radius from Arc Length and Angle

For part (b), use the formula \( s = r\theta \) again, where \( \theta = 128^{\circ} \) in radians is needed. Convert to radians:\[\theta = 128^{\circ} \times \frac{\pi}{180} \approx 2.234 \, \text{radians}\]Then solve for \( r \) using \( s = 0.14 \, \text{m} \) (converted from cm):\[0.14 = r \times 2.234 \Rightarrow r = \frac{0.14}{2.234} \approx 0.0627 \, \text{m} = 6.27 \, \text{cm}\]

Step 4: Find Arc Length from Given Angle and Radius

For part (c), use the formula for arc length \( s = r\theta \) with \( r = 1.50 \, \text{m} \) and \( \theta = 0.700 \, \text{radians} \):\[s = 1.50 \times 0.700 = 1.05 \, \text{m}\]

Key Concepts

Radians and Degrees Conversion

Understanding the conversion between radians and degrees is key in trigonometry as it allows us to switch between the different units of angular measurement. A circle encompasses 360 degrees, or equivalently, the angle measured in radians is equal to the circle's complete circumference divided by the radius.
To swap radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \), since one radian is approximately 57.3 degrees. Conversely, to convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \).

• Example: Convert 0.60 radians to degrees. Multiply by \( \frac{180}{\pi} \approx 34.38^\circ \).
• Convert 128 degrees to radians. Multiply by \( \frac{\pi}{180} \approx 2.234 \text{ radians} \).

This conversion is essential in solving problems related to circular motion, oscillations, and waves, where both radians and degrees may be employed interchangeably depending on the context.

Arc Length Calculation

Arc length is the distance along a section of the circumference of a circle. To find the arc length, use the formula \( s = r \theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians. By understanding these variables, you can quickly determine the length of an arc given sufficient information.

• If you know the angle in radians and radius, simply multiply them to get the arc length.
• If the angle is given in degrees, convert it to radians first before using the formula.
• Example: A circle with radius 1.50 m and angle 0.700 radians has an arc length of \( 1.50 \times 0.700 = 1.05 \text{ m} \).

This formula is particularly useful when working with circular objects in physics and engineering, such as wheels or disks, providing a straightforward way to link the angle subtended and the physical length of that section of the circle.

Circle Geometry

Circle geometry delves into various properties and relationships within a circle and provides fundamental principles used throughout mathematics and sciences. The circle's simple geometry makes it an essential element in various fields, including navigation, architecture, and even computer graphics.
A critical component of circle geometry is the relationship between the radius, diameter, and circumference of the circle. Knowing any two of these allows the calculation of the third:

• The radius is the distance from the center to a point on the circle.
• The diameter is twice the radius, spanning across the circle through its center.
• The circumference is the complete distance around the circle, calculated as \( 2\pi r \) or \( \pi d \).

Understanding these concepts also extends to analyzing arcs and sectors of the circle, which are segments defined by the center angle of the circle. The sectors and arcs are often used in finding cross-sectional areas or in analyzing sections of circular tracks.
This knowledge empowers solving exercises involving circular dimensions and angular relationships, equipping you with a toolkit for applications in both academic and practical contexts.