Problem 1 - University Physics with Modern Physics

Question

(a) What angle in radians is subtended by an arc 1.50 m long on the circumference of a circle of radius 2.50 m? What is this angle in degrees? (b) An arc 14.0 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 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 rad, 34.38°; (b) Radius = 6.27 cm; (c) Arc length = 1.05 m.

Step 1: Calculate Radian Measure from Arc Length and Radius

To find the radian measure \( \theta \) subtended by an arc on a circle, the formula is \( \theta = \frac{s}{r} \) where \( s \) is the arc length and \( r \) is the radius of the circle. For the given arc length of 1.50 m and radius of 2.50 m, apply the formula: \[ \theta = \frac{1.50}{2.50} = 0.60 \text{ radians} \]

Step 2: Convert Radians to Degrees

To convert an angle from radians to degrees, use the conversion factor \( 180^\circ / \pi \). Therefore, the angle in degrees is: \[ 0.60 \times \frac{180}{\pi} \approx 34.38^\circ \]

Step 3: Calculate Circle Radius from Arc Length and Angle

For the arc length \( s = 14.0 \text{ cm} \) and angle \( 128^\circ \), first convert the angle to radians: \[ 128^\circ \times \frac{\pi}{180} \approx 2.234 \text{ radians} \] Then, rearrange the formula \( \theta = \frac{s}{r} \) to solve for \( r \): \[ r = \frac{s}{\theta} = \frac{14.0}{2.234} \approx 6.27 \text{ cm} \]

Step 4: Calculate Arc Length from Angle and Radius

With a radius of 1.50 m and an angle of 0.700 rad, apply the formula \( s = r \times \theta \) to find the arc length: \[ s = 1.50 \times 0.700 = 1.05 \text{ m} \]

Key Concepts

Understanding Arc Length

Arc length is a portion of a circle's circumference. Imagine walking along the edge of a circle, the path you cover is the arc length. It's a central concept in geometry. Understanding the relationship between the arc length, circle radius, and angle is essential.

• Formula: To find an angle in radians using arc length, use the formula \( \theta = \frac{s}{r} \), where \( s \) is the arc length and \( r \) is the radius.
• Example: If you have an arc of 1.5 meters along a circle with a radius of 2.5 meters, apply the formula to get \( \theta = \frac{1.50}{2.50} = 0.60 \) radians.

For clarity, always remember that you need the arc and radius to calculate angles in radians. This helps in various applications, including converting into degrees.

Circle Radius and Its Importance

The radius is the distance from the center of the circle to any point along its edge. It provides a measure of the circle's size and is pivotal in calculating other geometric properties like arc length and angles.

• Determining Radius: When given an arc length and angle, you can find the circle radius by rearranging the formula \( \theta = \frac{s}{r} \) to \( r = \frac{s}{\theta} \).
• Example: Suppose an arc is 14 cm long and subtends an angle of 128 degrees. First, convert 128 degrees to radians (\( 128 \times \frac{\pi}{180} \approx 2.234 \) radians), then find the radius: \( r = \frac{14.0}{2.234} \approx 6.27 \) cm.

Using the radius, you can derive critical information about the circle, be it to find the circumference or, as in this context, solve for unknown properties from known ones.

Angle Conversion: Radians to Degrees

Converting angles from radians to degrees, and vice versa, is a basic yet crucial operation. Understanding this conversion helps in bridging problems that require visualizing angles.

• Conversion Formula: To convert radians to degrees, multiply the radian value by \( \frac{180}{\pi} \).
• Example: For an angle of 0.60 radians, you find degrees by: \( 0.60 \times \frac{180}{\pi} \approx 34.38^{\circ} \).

The ability to switch between radians and degrees allows ease of solving diverse problems in mathematics and physics. It’s vital when working through problems involving circular motion, such as calculating revolutions, where these conversions turn out handy.