Problem 1 - University Physics with Modern Physics

Question

(a) What angle in radians is subtended by an arc 1.50 \(\mathrm{m}\) long on the circumference of a circle of radius 2.50 \(\mathrm{m} ?\) What is this angle in degrees? (b) An are 14.0 \(\mathrm{cm}\) long 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) The angle is 0.60 radians or approx. 34.377 degrees. (b) The radius is approx. 0.0627 m. (c) The arc length is 1.05 m.

Step 1: Understanding the Relationship Between Arc Length and Angle

The formula that relates the arc length \(s\), angle \(\theta\) (in radians), and radius \(r\) of a circle is \(s = r \theta\). We will use this formula to solve part (a) and (c).

Step 2: Calculate the Angle in Radians (part a)

Using \(s = r \theta\), where \(s = 1.50 \text{ m}\) and \(r = 2.50 \text{ m}\), solve for \(\theta\) as follows: \(\theta = \frac{s}{r} = \frac{1.50}{2.50} = 0.60 \text{ radians}\).

Step 3: Convert Radians to Degrees (part a)

To convert radians to degrees, use the conversion factor \(1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees}\). Thus, \(0.60 \, \text{radians} = 0.60 \times \frac{180}{\pi} \approx 34.377 \text{ degrees}\).

Step 4: Solve for Radius (part b)

For part (b), use the formula \(s = r \theta\). First, convert the angle from degrees to radians: \(128^{\circ} = \frac{128\pi}{180} \approx 2.234\) radians. Then, rearrange the formula to find radius \(r\): \(r = \frac{s}{\theta}\). Here, \(s = 14.0 \text{ cm} = 0.14 \text{ m}\), which gives \(r = \frac{0.14}{2.234} \approx 0.0627 \text{ m} \).

Step 5: Calculate Arc Length (part c)

Use the formula \(s = r \theta\) again for part (c). Here, \(r = 1.50 \text{ m}\) and \(\theta = 0.700\) radians. Substitute the values to find \(s\): \(s = 1.50 \times 0.700 = 1.05 \text{ m}\).

Key Concepts

Understanding Angles in Radians

Radians measure angles based on the radius of a circle. While degrees are more commonly used in everyday life, radians provide a natural way to relate angles to circles. In mathematics, especially in calculus and physics, radians are preferred because they simplify the formulas and calculations.

One full circle is equivalent to an angle of 360 degrees, which can be represented in radians as the number of radii (from the center to the circumference) that fit around the circle. This is roughly \(2\pi\) radians for a full circle.

• A right angle is \(\pi/2\) radians.
• Straight angle is \(\pi\) radians.
• Full rotation is \(2\pi\) radians.

This means that understanding angles in radians helps us accurately describe angles with inherent circular properties.

Explaining Arc Length

The arc length is the measure of the distance along the curved line that makes up an arc, from one point on the circle to another. When you know the angle in radians (\(\theta\)) and the radius (\(r\)) of the circle, you can easily find the arc length (\(s\)) using the formula: \(s = r\theta\).

For example, if you have an angle of \(0.700\) radians in a circle with a radius of \(1.50\) m, the arc length would be calculated as \((1.50 \, \text{m}) \times 0.700 = 1.05 \, \text{m}\).

• This formula directly relates the central angle in radians to the length of the arc.
• Arc length tells us how much of the circle's circumference is covered by the arc.

Understanding arc length is essential for solving problems related to the circumference of a circle and paths along curved surfaces.

Conversion Between Radians and Degrees

To convert an angle from radians to degrees, or vice versa, it's necessary to understand their relationship. The conversion is straightforward:

• 1 radian = \(\frac{180}{\pi}\) degrees
• 1 degree = \(\frac{\pi}{180}\) radians

So, to convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\), and for degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\).

For example, if an angle is \(0.60\) radians, converting it to degrees involves the calculation \(0.60 \times \frac{180}{\pi} \approx 34.38\) degrees.
This understanding makes it easier to switch between different units of angle measurement, ensuring precise calculations depending on the context.

Radius of a Circle: Concept and Importance

The radius of a circle is the distance from the center of the circle to any point on its circumference. It's a crucial part of circle geometry as it defines the size of the circle and is a variable in calculating key aspects like the circumference, area, and arc length.

The radius is used in the formula for arc length (\(s = r\theta\)), making it essential in determining how long an arc is when combined with a given angle in radians.

• Radius influences the circle's circumference: \(C = 2\pi r\).
• It determines circular area: \(A = \pi r^2\).
• It's fundamental in trigonometry and geometry calculations.

Knowing the radius allows for precise calculations in various mathematical applications, providing a basis for understanding and solving circle-related problems effectively.