Problem 25
Question
Where needed, assume the earth to be a sphere with a radius of 3960 mi. Actually, the distance from pole to pole is about 27 mi less than the diameter at the equator. A 1.25 -m-long pendulum swings \(5.75^{\circ}\) on each side of the vertical. Find the length of arc traveled by the end of the pendulum.
Step-by-Step Solution
Verified Answer
The length of arc traveled by the end of the pendulum is approximately \(0.25\pi \) m.
1Step 1: Calculate the Circumference of the Circle
The pendulum swings in a circular arc. To find the arc length, first calculate the circumference of the circle formed by the full swing of the pendulum using the formula for the circumference of a circle, which is given by \( C = 2\pi r \), where \( r \) is the length of the pendulum. Here, \( r = 1.25 \) m. So calculate \( C = 2\pi \times 1.25 \) m.
2Step 2: Determine the Portion of the Circle for the Swing
The pendulum swings through an angle of \(5.75^\circ\) on each side of the vertical, making the total angle of the swing \(2 \times 5.75^\circ = 11.5^\circ\). To find out what fraction of the circle this angle represents, divide the angle of the swing by the total angle in a circle, which is 360 degrees. So the fraction is \(\frac{11.5}{360}\).
3Step 3: Calculate the Arc Length Traveled
The arc length \( L \) is a portion of the circumference. It corresponds to the fraction of the 360-degree circle that the pendulum swings through. Use the formula \( L = \frac{\text{angle}}{360} \times C \) to find the arc length. Plug in the angle of \(11.5^\circ\) and the circumference calculated in Step 1 to get \( L = \frac{11.5}{360} \times 2\pi \times 1.25 \) m.
Key Concepts
Circumference of a CircleCentral Angle of a CircleRadius and Arc Length Relationship
Circumference of a Circle
Understanding the circumference of a circle is a fundamental aspect of geometry that is widely applied in various problems, including calculating arc lengths. The circumference can be thought of as the perimeter of the circle, which is the distance around it. Imagine wrapping a piece of string around a circular object; the length of that string would be the circumference.
To calculate the circumference, a simple formula is used: \[ C = 2\text{{\textpi}}r \], where \( C \) stands for the circumference, \( \text{{\textpi}} \) (approximately 3.14159) is a constant representing the ratio of the circumference of any circle to its diameter, and \( r \) is the radius of the circle, or the distance from the center of the circle to any point on its edge. This formula is a fundamental tool in problems involving circular motion or geometry, such as the one involving the pendulum swing.
For our pendulum with a length of 1.25 meters, which acts as the radius, we can use this formula to calculate the total distance around the path the pendulum would take if it were to make a complete circle.
To calculate the circumference, a simple formula is used: \[ C = 2\text{{\textpi}}r \], where \( C \) stands for the circumference, \( \text{{\textpi}} \) (approximately 3.14159) is a constant representing the ratio of the circumference of any circle to its diameter, and \( r \) is the radius of the circle, or the distance from the center of the circle to any point on its edge. This formula is a fundamental tool in problems involving circular motion or geometry, such as the one involving the pendulum swing.
For our pendulum with a length of 1.25 meters, which acts as the radius, we can use this formula to calculate the total distance around the path the pendulum would take if it were to make a complete circle.
Central Angle of a Circle
The central angle of a circle is another critical concept to grasp when dealing with arc lengths. The central angle is formed by two radii and the arc that lies between them, essentially creating a 'slice' of the circle. The size of this angle is measured in degrees or radians, with one complete circle equating to \(360^{\text{{\textdegree}}}\) or \(2\text{{\textpi}}\) radians.
The relationship between the central angle and the total angle in a circle plays a vital role in determining how large a portion of the circumference the arc length represents. If you know the degree measure of the central angle, you can easily determine what fraction of the circle's circumference the arc length is by dividing the central angle by \(360^{\text{{\textdegree}}}\). For instance, in the pendulum problem, the pendulum swings \(5.75^{\text{{\textdegree}}}\) on each side of the vertical, hence the total central angle of the swing is \(11.5^{\text{{\textdegree}}}\). This angle helps us to partition the circle and find the specific length of the arc we're interested in.
The relationship between the central angle and the total angle in a circle plays a vital role in determining how large a portion of the circumference the arc length represents. If you know the degree measure of the central angle, you can easily determine what fraction of the circle's circumference the arc length is by dividing the central angle by \(360^{\text{{\textdegree}}}\). For instance, in the pendulum problem, the pendulum swings \(5.75^{\text{{\textdegree}}}\) on each side of the vertical, hence the total central angle of the swing is \(11.5^{\text{{\textdegree}}}\). This angle helps us to partition the circle and find the specific length of the arc we're interested in.
Radius and Arc Length Relationship
The relationship between radius and arc length is vital in determining the distance traveled by an object moving along a circular path. An arc length is a measure of the distance along a section of the perimeter of a circle. It is directly proportional to the radius of the circle and the central angle that subtends the arc.
The formula to find the arc length \( L \) based on these two components is given by \[ L = \frac{\text{{angle}}}{360^\text{{\textdegree}}} \times C \], where \( C \) is the circumference of the circle, and 'angle' represents the central angle in degrees. Given a fixed radius, the arc length increases as the central angle increases since a larger angle will cover more of the circle's circumference.
In the context of our pendulum problem, the length of the pendulum acts as the radius. Using the formula, the arc length traveled by the end of the pendulum can be obtained by knowing the central angle of its swing and the circumference of the circle that would be formed if the pendulum completed a full swing. Through this relationship, we can accurately calculate the length of the arc without measuring it directly, simply by using the pendulum’s full length and its swing angle.
The formula to find the arc length \( L \) based on these two components is given by \[ L = \frac{\text{{angle}}}{360^\text{{\textdegree}}} \times C \], where \( C \) is the circumference of the circle, and 'angle' represents the central angle in degrees. Given a fixed radius, the arc length increases as the central angle increases since a larger angle will cover more of the circle's circumference.
In the context of our pendulum problem, the length of the pendulum acts as the radius. Using the formula, the arc length traveled by the end of the pendulum can be obtained by knowing the central angle of its swing and the circumference of the circle that would be formed if the pendulum completed a full swing. Through this relationship, we can accurately calculate the length of the arc without measuring it directly, simply by using the pendulum’s full length and its swing angle.
Other exercises in this chapter
Problem 24
Convert each angle given in degrees to radian measure in terms of \(\pi\) $$105^{\circ}$$
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Convert each angle given in degrees to radian measure in terms of \(\pi\) $$78^{\circ}$$
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Convert each angle given in degrees to radian measure in terms of \(\pi\) $$305^{\circ}$$
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