Question

Think \& Calculate As Tony the fisherman reels in a "big one," he turns the spool on his fishing reel at the rate of \(3.0\) complete revolutions every second. See Figure 8.26. (a) If the radius of the reel is \(3.7 \mathrm{~cm}\), what is the linear speed of the fishing line as it is reeled in? (b) How would your answer to part (a) change if the radius of the reel were doubled?

Short answer

The linear speed for a 3.7 cm radius is approximately 69.7 cm/s. If the radius is doubled, the speed would be approximately 139.4 cm/s.

Step-by-step solution

Identify Given Values

We know from the problem statement that the spool on the fishing reel makes 3.0 complete revolutions per second and the radius of the reel is 3.7 cm. We need to find the linear speed of the fishing line.

Convert Revolutions to Radians

One complete revolution is equal to \(2\pi\) radians. Therefore, the angular speed \(\omega\) in radians per second is given by \(\omega = 3.0 \times 2\pi = 6\pi\, \text{radians/second}\).

Calculate Linear Speed Using Angular Speed

The linear speed \(v\) can be calculated from the angular speed using the formula \(v = r \times \omega\), where \(r\) is the radius of the reel. Substituting the values, we get \(v = 3.7 \times 6\pi\).

Calculate Result for Original Radius

Substitute \(r = 3.7\, \text{cm}\) into the equation: \(v = 3.7 \times 6\pi\, \text{cm/s}\). Calculate this to find \(v \approx 69.7\, \text{cm/s}\).

Consider Change in Radius (Part b)

If the radius is doubled, the new radius is \(2 \times 3.7 = 7.4\, \text{cm}\). Use the same formula: \(v = 7.4 \times 6\pi\, \text{cm/s}\).

Calculate Result for Doubled Radius

Substitute the new \(r = 7.4\) cm into the equation: \(v = 7.4 \times 6\pi\, \text{cm/s}\). Calculate this to find \(v \approx 139.4\, \text{cm/s}\).

Key concepts

Angular Speed

Angular speed is a measure of how quickly an object rotates or revolves relative to another point. It is expressed in terms of angles covered per unit time. In this exercise, we're dealing with a spool on a fishing reel that turns at 3 revolutions per second. To understand angular speed, think of it like the thrill of a merry-go-round spinning. The faster it spins, the greater the angular speed.

• Angular speed (\( \omega \)) is typically expressed in radians per second (\( \text{rad/s} \)).
• One complete revolution corresponds to \(2\pi\) radians.
• For this exercise: \( \omega = 3.0 \times 2\pi = 6\pi \text{ rad/s} \)

Hence, angular speed helps define how fast the spool is turning, which is crucial to determine the linear speed of the fishing line.

Revolutions per Second

Revolutions per second indicate how many complete turns an object makes in one second. It's a straightforward concept often used to describe rotational motion, like a car tire spinning or a ceiling fan rotating.
Here's why revolutions per second are essential in the fishing reel example:

• We know the spool completes 3 revolutions in one second.
• Each revolution represents a full 360-degree turn by the spool.
• This base measurement is converted into radians for further calculations—specifically to find angular speed.

By acknowledging the number of revolutions per second, you can transition to more complex calculations, like angular speed, and eventually determine linear speed.

Radians Conversion

Radians conversion may sound complex, but it's a simple way to express angles in terms of \( \pi \), making them easier to calculate and use in formulas. In the exercise, we convert the spool's revolutions into radians.

• One revolution = \(2\pi\) radians.
• Converting revolutions to radians simplifies mathematical applications like calculating angular speed.
• For example, 3 revolutions per second become \(3 \times 2\pi = 6\pi\)

Radians are advantageous because they directly relate an angle's length or curvature to the unit circle, paving the way for seamless calculations involving trigonometry and physics.

Effect of Radius on Speed

The radius of a circular object significantly influences its speed variable outcomes when concerning rotational movement. Whether you’re reeling in a big fish or racing a bicycle, the radius matters.

• Linear speed (\( v \)) directly hinges on radius and angular speed, encapsulated in the formula \( v = r \times \omega \).
• Given \( r = 3.7 \text{ cm}\) in the original setup, multiplying by the angular speed \( 6\pi \text{ rad/s}\) gives a linear speed of about \( 69.7 \text{ cm/s}\).
• If the radius doubles, say to \( 7.4 \text{ cm}\), the linear speed also doubles, reaching \( 139.4 \text{ cm/s}\).

Doubling the radius basically gives you twice the linear speed, demonstrating how changes in radius size can drastically alter speed outcomes, highlighting the pivotal role of radius in rotational motion dynamics.