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Chapter 9: Problem 1

(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?

Short Answer

Expert verified
(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 by step solution

01

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).
02

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: \(\theta = \frac{s}{r} = \frac{1.50}{2.50} = 0.60 \text{ radians}\).
03

Convert Radians to Degrees (part a)

To convert radians to degrees, use the conversion factor \(1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees}\). Thus, \(0.60 \, \text{radians} = 0.60 \times \frac{180}{\pi} \approx 34.377 \text{ degrees}\).
04

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 \(r = \frac{0.14}{2.234} \approx 0.0627 \text{ m} \).
05

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 \(s\): \(s = 1.50 \times 0.700 = 1.05 \text{ m}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A uniform \(3.00-\mathrm{kg}\) rope 24.0 \(\mathrm{m}\) lies on the ground at the mtop of a vertical cliff. A mountain climber at the top lets down half of it to help his partner climb up the cliff. What was the change in potential energy of the rope during this maneuver?

The moment of inertia of a sphere with uniform density about an axis through its center is \(\frac{2}{5} M R^{2}=0.400 M R^{2} .\) Satellite observations show that the earth's moment of inertia is 0.3308\(M R^{2}\) . Geophysical data suggest the earth consists of flve main regions: the inner core \((r=0 \text { to } r=1220 \mathrm{kn})\) of average density 12, \(900 \mathrm{kg} / \mathrm{m}^{3},\) the outer core \((r=1220 \mathrm{kin} \text { to } r=3480 \mathrm{kin})\) of average density \(10,900 \mathrm{kg} / \mathrm{m}^{3},\) the lower mantle \((r=3480 \mathrm{kn} \text { to }\) \(r=5700 \mathrm{kin}\) of average density 4900 \(\mathrm{kg} / \mathrm{m}^{3}\) , the upper mantle \((r=5700 \mathrm{kn} \text { to } r=6350 \mathrm{kn})\) of average density 3600 \(\mathrm{kg} / \mathrm{m}^{3}\) and the outer crust and oceans \((r=6350 \mathrm{km} \text { to } r=6370 \mathrm{kn})\) of average density 2400 \(\mathrm{kg} / \mathrm{m}^{3}\) . (a) Show that the moment of inertia about a diameter of a uniform spherical shell of inner radius \(R_{1}\) . outer radius \(R_{2}\) , and density \(\rho\) is \(I=\rho(8 \pi / 15)\left(R_{2}^{5}-R_{1}^{5}\right) .\) (Hint: Form the shell by superposition of a sphere of density \(\rho\) and a smaller sphere of density \(-\rho .\) (b) Check the given data by using them to calculate the mass of the earth. (c) Use the given data to calculate the earth's moment of inertia in terms of \(M R^{2}\) .

You need to design an industrial turntable that is 60.0 \(\mathrm{cm}\) in diameter and has a kinetic energy of 0.250 \(\mathrm{J}\) when turning at 45.0 \(\mathrm{rpm}(\mathrm{rev} / \mathrm{min})\) . (a) What must be the moment of inertia of the turntable about the rotation axis? (b) If your workshop makes this turntable in the shape of a uniform solid disk, what must be its mass?

A thin, uniform rod is bent into a square of side length \(a\) . If the total mass is \(M\) , find the moment of inertia about an axis through the center and perpendicular to the plane of the square. (Hint: Use the parallel-axis theorem.)

A compound disk of outside diameter 140.0 \(\mathrm{cm}\) is made up of a uniform solid disk of radius 50.0 \(\mathrm{cm}\) and area density 3.00 \(\mathrm{g} / \mathrm{cm}^{2}\) surrounded by a concentric ring of inner radius 50.0 \(\mathrm{cm}\) , outer radius \(70.0 \mathrm{cm},\) and area density 2.00 \(\mathrm{g} / \mathrm{cm}^{2} .\) Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.
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