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Chapter 7: Problem 2

A wheel has a radius of \(4.1 \mathrm{~m}\). How far (path length) does a point on the circumference travel if the wheel is rotated through angles of \(30^{\circ}, 30 \mathrm{rad}\), and 30 rev, respectively?

Short Answer

Expert verified
When the wheel is rotated through angles of \(30^{\circ}\), \(30 rad\), and \(30 rev\), the respective path lengths are \(4.1 m \cdot 30^{\circ} \cdot \frac{\pi}{180} rad = 7.15 m, 4.1 m \cdot 30 rad = 123 m, 4.1 m \cdot 30 rev = 4.1 m \cdot 30 \cdot 2\pi rad = 772.3 m\).

Step by step solution

01

Calculate path length for \(30^{\circ}\) rotation

First, note that the given angle is in degrees. So, we need to convert it into radians. The conversion ratio is \(1 rad = \frac{180}{\pi} degrees\), thus \(30^{\circ} = 30 \cdot \frac{\pi}{180} rad\). Then use the formula \(s = r \cdot \theta\) to calculate the path length. Substituting the values, we have \(s = 4.1 m \cdot 30^{\circ} = 4.1 m \cdot 30 \cdot \frac{\pi}{180} rad\). Solve to get the path length.
02

Calculate path length for \(30 rad\) rotation

Now, the angle is already given in radians. Thus we can directly use the formula \(s = r \cdot \theta\). Substituting the values, we have \(s = 4.1 m \cdot 30 rad\). Solve to get the path length.
03

Calculate path length for 30 rev rotation

When the angle of rotation is given in revolutions, knowing that one revolution is \(2\pi\) radians, you replace the revolutions with this equivalent amount of radians. Thus, in our formula \(s = r \cdot \theta\) , the angle becomes \(30 \cdot 2\pi rad\). Substituting the values, we have \(s = 4.1 m \cdot 30 \cdot 2\pi rad\). Solve to get the path length.

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Key Concepts

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

Conversion of Angles
When working with angles in geometry and trigonometry, it's essential to understand how to convert between different units of measurement. Angles can be expressed in degrees, radians, or revolutions, and being able to switch between these units can be very helpful.
  • **Degrees** are a common measure, where a full circle is 360 degrees. To convert degrees to radians, use the formula: \[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]
  • **Radians** are the standard unit in calculus and many other branches of mathematics because they relate directly to the properties of the unit circle. In radians, a full circle is \(2\pi\). The conversion from radians to degrees is the inverse:\[\text{degrees} = \text{radians} \times \frac{180}{\pi}\]
  • **Revolutions** indicate how many times a circle is completely rotated. One revolution is equivalent to \(360^{\circ}\) or \(2\pi\) radians.
By mastering these conversions, you'll have a much easier time solving problems that involve angles and rotations.
Radian Measure
Radians offer a natural way to measure angles, particularly when dealing with trigonometry and calculus. They are based on the radius of a circle, which makes them incredibly intuitive for calculating path lengths and arc sizes.
Here's how radians work:
  • **Definition**: One radian is the angle created when the radius wraps around the circle's edge an arc length equal to the radius itself.
  • **Circle Relation**: A full circle (360 degrees) is \(2\pi\) radians. A half circle (180 degrees) is \(\pi\) radians. This direct relation with the circle is why radians are so frequently used in mathematics.
  • **Use in Calculations**: Because radians link directly to the properties of the circle, they simplify many mathematical formulas, such as the calculation of arc lengths, surface areas, and solving various integrals.
Understanding and utilizing radian measure will simplify and enhance your mathematical problem-solving skills, especially for angle-related tasks.
Arc Length Formula
The arc length formula is a powerful tool used to find the distance traveled along the edge of a circle or arc. This formula is especially useful in the context of circular motion, such as wheels or gears.
Here's a breakdown of the process:
  • **The Formula**: The arc length \(s\) of a circle is calculated using the formula:\[s = r \cdot \theta\]where \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.
  • **Why Use Radians**: The formula only works when \(\theta\) is in radians. Radians provide a direct proportionality between the angle and the arc length.
  • **Applications**: This formula is particularly useful in physics and engineering when calculating the distance covered by rotating wheels, pulleys, and other circular objects. It can also be used in creating sectors of circles in geometric problems.
Mastering the arc length formula can help solve complex problems involving rotations and circles, providing precise measurements and insights into geometric relationships.

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Most popular questions from this chapter

A \(600-\mathrm{kg}\) satellite is in a circular orbit about Earth at a height above Earth equal to Earth's mean radius. Find (a) the satellite's orbital speed, (b) the period of its revolution, and (c) the gravitational force acting on it.

ecp Because of Earth's rotation about its axis, a point on the equator has a centripetal acceleration of \(0.0340 \mathrm{~m} / \mathrm{s}^{2}\) whereas a point at the poles has no centripetal acceleration. (a) Show that, at the equator, the gravitational force on an object (the object's true weight) must exceed the object's apparent weight. (b) What are the apparent weights of a \(75.0\) -kg person at the equator and at the poles? (Assume Earth is a uniform sphere and take \(g=\) \(\left.9.800 \mathrm{~m} / \mathrm{s}^{2} .\right)\)

A coordinate system (in meters) is constructed on the surface of a pool table, and three objects are placed on the table as follows: a \(2.0\) -kg object at the origin of the coordinate system, a \(3.0\) -kg object at \((0,2.0)\), and a \(4.0\) -kg object at \((4.0,0)\). Find the resultant gravitational force exerted by the other two objects on the object at the origin.

Suppose a 1800 -kg car passes over a bump in a roadway that follows the arc of a circle of radius \(20.4 \mathrm{~m}\), as in Figure \(\mathrm{P} 7.65 .\) (a) What force does the road exert on the car as the car passes the highest point of the bump if the car travels at \(8.94 \mathrm{~m} / \mathrm{s} ?\) (b) What is the maximum speed the car can have without losing contact with the road as it passes this highest point?

A dentist's drill starts from rest. After \(3.20 \mathrm{~s}\) of constant angular acceleration, it turns at a rate of \(2.51 \times 10^{4} \mathrm{rev} / \mathrm{min}\). (a) Find the drill's angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.
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