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Chapter 8: Problem 31

Refer to Interactive Solution \(\underline{8.31}\) at to review a model for solving this problem. The take-up reel of a cassette tape has an average radius of \(1.4 \mathrm{~cm} .\) Find the length of tape (in meters) that passes around the reel in \(13 \mathrm{~s}\) when the reel rotates at an average angular speed of \(3.4 \mathrm{rad} / \mathrm{s}\)

Short Answer

Expert verified
0.6188 meters of tape pass around the reel in 13 seconds.

Step by step solution

01

Calculate the Circumference of the Reel

The circumference of a circle is given by the formula \( C = 2\pi r \), where \( r \) is the radius. Given that the radius of the take-up reel is \( 1.4 \) cm, we can calculate:\[C = 2\pi \times 1.4 = 2.8\pi \text{ cm}\]
02

Determine the Total Angle Rotated

To find the total angle rotated in radians, multiply the angular speed \( \omega = 3.4 \) rad/s by the time \( t = 13 \) seconds:\[\theta = \omega \times t = 3.4 \times 13 = 44.2 \text{ radians}\]
03

Compute the Length of Tape

The length of tape passing around the reel is the product of the total angle rotated and the radius. This is given by the formula \( \text{Length} = \theta \times r \). Substituting the values from the previous steps:\[\text{Length} = 44.2 \times 1.4 = 61.88 \text{ cm}\]
04

Convert Length from Centimeters to Meters

Since the problem asks for the length in meters, we need to convert the length of the tape from centimeters to meters. We do this by dividing by 100:\[\text{Length in meters} = \frac{61.88}{100} = 0.6188 \text{ meters}\]

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Key Concepts

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

Circumference Calculation
The circumference of a circle is an essential concept in geometry, specifically when dealing with circular motion. To find the circumference, use the formula \( C = 2\pi r \), where \( r \) is the radius. In practical terms, the circumference represents the distance around the circle. For example, with a radius of \( 1.4 \text{ cm} \), we calculate:
\[C = 2\pi \times 1.4 = 2.8\pi \text{ cm}\]
This calculation helps us understand the distance the tape would cover in one complete rotation around the reel. The use of \( \pi \) is crucial here as it directly links to the properties of circles.
Angular Speed
Angular speed is a measure of how quickly an object rotates. It is usually denoted by \( \omega \) and measured in radians per second \( \text{rad/s} \). Angular speed tells us how fast the angle is changing over time. This concept is important in many applications, such as understanding how fast a reel spins.
In this exercise, the reel rotates at an average angular speed of \( 3.4 \text{ rad/s} \). It means that every second, the reel rotates through \( 3.4 \) radians. This measurement is vital to calculate how much of the tape is used in a set amount of time.
Length Conversion
Length conversion is about changing measurements from one unit to another, making them easier to understand or compare. Here, we convert the length of tape from centimeters to meters. This is important because it allows us to express the tape length in a more universally understood unit.
Since there are 100 centimeters in a meter, the conversion is straightforward:
\[\text{Length in meters} = \frac{61.88}{100} = 0.6188 \text{ meters}\]
This conversion ensures clarity, especially in scientific contexts where meters are a standard unit of measure.
Total Angle in Radians
The total angle in radians gives us an idea of how much rotation has occurred in a given time. It's calculated by multiplying the angular speed \( \omega \) by the time \( t \). Radians are a natural unit of angular measure often used in mathematics and physics.
Here鈥檚 how you calculate the total angle:
\[\theta = \omega \times t = 3.4 \times 13 = 44.2 \text{ radians}\]
This means the reel completed 44.2 radians of rotation, determining how much tape was used in 13 seconds. Understanding radians helps simplify many calculations involving circular motion.

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Most popular questions from this chapter

One type of slingshot can be made from a length of rope and a leather pocket for holding the stone. The stone can be thrown by whirling it rapidly in a horizontal circle and releasing it at the right moment. Such a slingshot is used to throw a stone from the edge of a cliff, the point of release being \(20.0 \mathrm{~m}\) above the base of the cliff. The stone lands on the ground below the cliff at a point \(X .\) The horizontal distance of point \(X\) from the base of the cliff (directly beneath the point of release) is thirty times the radius of the circle on which the stone is whirled. Determine the angular speed of the stone at the moment of release.

The front and rear sprockets on a bicycle have radii of 9.00 and \(5.10 \mathrm{~cm}\), respectively The angular speed of the front sprocket is \(9.40 \mathrm{rad} / \mathrm{s}\). Determine (a) the linear speed (in \(\mathrm{cm} / \mathrm{s}\) ) of the chain as it moves between the sprockets and \((\mathrm{b})\) the centripetal acceleration (in \(\mathrm{cm} / \mathrm{s}^{2}\) ) of the chain as it passes around the rear sprocket.

At the local swimming hole, a favorite trick is to run horizontally off a cliff that is \(8.3 \mathrm{~m}\) above the water. One diver runs off the edge of the cliff, tucks into a "ball", and rotates on the way down with an average angular speed of 1.6 rev/s. Ignore air resistance and determine the number of revolutions she makes while on the way down.

A baton twirler throws a spinning baton directly upward. As it goes up and returns to the twirler's hand, the baton turns through four revolutions. Ignoring air resistance and assuming that the average angular speed of the baton is 1.80 rev/s, determine the height to which the center of the baton travels above the point of release.

An electric drill starts from rest and rotates with a constant angular acceleration. After the drill has rotated through a certain angle, the magnitude of the centripetal acceleration of a point on the drill is twice the magnitude of the tangential acceleration. What is the angle?
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