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Chapter 9: Problem 1

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 arc 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) 0.60 radians, 34.38掳; b) Radius is 6.28 cm; c) Arc length 1.05 m.

Step by step solution

01

Convert arc length to radians (Part a)

The formula to find the angle in radians \( \theta \) subtended by an arc is \( \theta = \frac{s}{r} \) where \( s \) is the arc length and \( r \) is the radius. So here, \( \theta = \frac{1.50}{2.50} \). Calculate to get \( \theta = 0.60 \) radians.
02

Convert radians to degrees (Part a)

To convert radians to degrees, use the conversion \( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \).So, \( \theta \text{ in degrees} = 0.60 \times \frac{180}{\pi} \approx 34.38^{\circ} \).
03

Calculate radius using arc length and angle (Part b)

For the second part, use the formula again \( s = r \theta \) but rearranged to solve for \( r \), i.e., \( r = \frac{s}{\theta} \).First, convert the angle from degrees to radians: \( 128^{\circ} = \frac{128 \pi}{180} \approx 2.23 \text{ radians} \).Then, \( r = \frac{14.0 \text{ cm}}{2.23} \approx 6.28 \text{ cm} \).
04

Calculate arc length (Part c)

For the third part, use the arc length formula again: \( s = r \times \theta \).Given that \( r = 1.50 \text{ m} \) and \( \theta = 0.700 \text{ rad} \),calculate \( s = 1.50 \times 0.700 = 1.05 \text{ m} \).

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Key Concepts

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

Arc Length
Arc length is an essential concept in circle geometry. It represents the distance along the curved edge of a circle. To find the arc length, you use the formula:
  • \( s = r \theta \)
where \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the angle in radians.

If you know the angle in degrees, convert it to radians first, because the formula requires the angle to be in radians. Understanding how to properly use this formula helps identify the portion of the circle described by an arc, which can be very useful in practical problems involving circles, like tracks or wheels.
Angle Conversion
In circle geometry, converting angles between degrees and radians is a frequent task. Knowing the relationship between these two units is crucial. The following formulas will help:
  • From radians to degrees: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)
  • From degrees to radians: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)
Each complete circle has \(360^{\circ}\) or \(2\pi\) radians. This implies a direct conversion factor between degrees and radians.

Remember: \(180^{\circ} = \pi\) radians. Having a firm grasp on these conversions enables effective problem solving, especially in varied contexts where angles need to be expressed in different units.
Radians and Degrees
Radians and degrees are two units of measuring angles. Degrees are more familiar in everyday contexts and are marked by the symbol \(^{\circ}\). Radians, on the other hand, offer mathematical simplicity within calculations, especially for angles in circles.

A radian is based on the radius of the circle. One radian is the angle created when the arc length is equal to the radius. This naturally leads to the fact that \(2\pi\) radians make a full circle, equating to \(360^{\circ}\).
  • A quarter circle has \(90^{\circ}\) or \(\frac{\pi}{2}\) radians.
  • Half a circle equals \(180^{\circ}\) or \(\pi\) radians.
Understanding radians is vital in fields like engineering and physics because the trigonometric functions relating to radians translate seamlessly into other mathematical contexts.
Geometry Formulas
Circle geometry involves several key formulas that are foundational for solving problems involving circles. Not just limited to arc lengths and angles, these formulas extend to other properties of the circle.
  • Area of a circle: \( A = \pi r^2 \)
  • Circumference of a circle: \( C = 2\pi r \)
  • Sector area: \( A = \frac{1}{2} r^2 \theta \) for \( \theta \) in radians
These formulas create a toolkit for exploring the properties of circles in depth.

By practicing their application, you become equipped not only to solve textbook problems but also to model real-world situations with efficiency and accuracy.

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Most popular questions from this chapter

At \(t=0\) a grinding wheel has an angular velocity of 24.0 rad/s. It has a constant angular acceleration of 30.0 rad/s \(^{2}\) until a circuit breaker trips at \(t=2.00\) s. From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. (a) Through what total angle did the wheel turn between \(t=0\) and the time it stopped? (b) At what time did it stop? (c) What was its acceleration as it slowed down?

A roller in a printing press turns through an angle \(\theta(t)\) given by \(\theta(t)=y t^{2}-\beta t^{3},\) where \(\gamma=3.20 \mathrm{rad} / \mathrm{s}^{2}\) and \(\beta=0.500 \mathrm{rad} / \mathrm{s}^{3}\) (a) Calculate the angular velocity of the roller as a function of time. (b) Calculate the angular acceleration of the roller as a function of time. (c) What is the maximum positive angular velocity, and at what value of \(t\) does it occur?

A flywheel has angular acceleration \(\alpha_{z}(t)=$$8.60 \mathrm{rad} / \mathrm{s}^{2}-\left(2.30 \mathrm{rad} / \mathrm{s}^{3}\right) t,\) where counterclockwise rotation is positive. (a) If the flywheel is at rest at \(t=0,\) what is its angular velocity at 5.00 s? (b) Through what angle (in radians) does the flywheel turn in the time interval from \(t=0\) to \(t=5.00\) s?

The flywheel of a gasoline engine is required to give up 500 \(\mathrm{J}\) of kinetic energy while its angular velocity decreases from 650 \(\mathrm{rev} / \mathrm{min}\) to 520 \(\mathrm{rev} / \mathrm{min}\) . What moment of inertia is required?

On a compact disc (CD), music is coded in a pattern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a \(\mathrm{CD}\) player, the track is scanned at a constant linear speed of \(v=1.25 \mathrm{m} / \mathrm{s}\) Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the CD is played. (See Exercise \(9.20 .\) ) Let's see what angular acceleration is required to keep \(v\) constant. The equation of a spiral is \(r(\theta)=r_{0}+\beta \theta\) , where \(r_{0}\) is the radius of the spiral at \(\theta=0\) and \(\beta\) is a constant. On a \(\mathrm{CD}, r_{0}\) is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, \(\beta\) must be positive so that \(r\) increases as the disc turns and \(\theta\) increases. (a) When the disc rotates through a small angle \(d \theta\) , the distance scanned along the track is \(d s=r d \theta .\) Using the above expression for \(r(\theta),\) integrate \(d s\) to find the total distance \(s\) scanned along the track as a function of the total angle \(\theta\) through which the disc has rotated. (b) since the track is scanned at a constant linear speed \(v,\) the distance \(s\) found in part (a) is equal to \(v t\) . Use this to find \(\theta\) as a function of time. There will be two solutions for \(\theta\) ; choose the positive one, and explain why this is the solution to choose. (c) Use your expression for \(\theta(t)\) to find the angular velocity \(\omega_{z}\) and the angular acceleration \(\alpha_{z}\) as functions of time. Is \(\alpha_{z}\) constant? (d) On a CD, the inner radius of the track is 25.0 \(\mathrm{mm}\) , the track radius increases by 1.55\(\mu \mathrm{m}\) per revolution, and the playing time is 74.0 min. Find the values of \(r_{0}\) and \(\beta,\) and find the total number of revolutions made during the playing time. (e) Using your results from parts (c) and (d), make graphs of \(\omega_{z}\) \((\) in \(\operatorname{rad} / \mathrm{s})\) versus \(t\) and \(\alpha_{z}\) \((\) in \(\operatorname{rad} / \mathrm{s}^2)\) versus \(t\) between \(t=0\) and \(t=74.0\) min.
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