Question

In a tape recorder, the magnetic tape moves at a constant linear speed of \(5.6 \mathrm{~cm} / \mathrm{s}\). To maintain this constant linear speed, the angular speed of the driving spool (the take-up spool) has to change accordingly. a) What is the angular speed of the take-up spool when it is empty, with radius \(r_{1}=0.80 \mathrm{~cm} ?\) b) What is the angular speed when the spool is full, with radius \(r_{2}=2.20 \mathrm{~cm} ?\) c) If the total length of the tape is \(100.80 \mathrm{~m}\), what is the average angular acceleration of the take-up spool while the tape is being played?

Short answer

Answer: The average angular acceleration of the take-up spool while the tape is being played is approximately -0.247 rad/s².

Step-by-step solution

Initial Angular Speed of the Empty Spool (Scenario A)

The magnetic tape moves at a constant linear speed of 5.6 cm/s, and the initial radius of the empty spool is 0.80 cm. To find the angular speed of the empty spool, we will use the formula: \(v = \omega \times r\) Rearranging the formula to solve for the angular speed, \(\omega = \frac{v}{r}\) Now, substitute the given values, \(\omega = \frac{5.6}{0.80}\) Calculating the result, \(\omega = 7 \; \mathrm{rad/s}\) So, the initial angular speed of the empty spool is 7 rad/s.

Angular Speed of the Full Spool (Scenario B)

Now, we need to find the angular speed of the spool when it is full, with a radius of 2.20 cm. Using the formula from step 1, we have: \(v = \omega \times r\) Rearranging the formula to solve for the angular speed, \(\omega = \frac{v}{r}\) Substituting the given values, \(\omega = \frac{5.6}{2.20}\) Calculating the result, \(\omega \approx 2.55\; \mathrm{rad/s}\) So, the angular speed of the full spool is about 2.55 rad/s.

Calculate the Average Angular Acceleration

Now, we will find the average angular acceleration while the tape is being played. Given the total length of the tape (100.80 m), we can calculate the time taken to play the entire tape, as we have the linear speed: Time taken (Δt) = Total length of the tape / Linear speed \(\Delta t = \frac{100.80}{5.6}\) Calculating the result, \(\Delta t \approx 18.00 \; \mathrm{s}\) We have the initial and final angular speeds from steps 1 and 2, so we can now find the average angular acceleration using the formula: \(\alpha = \frac{\Delta \omega}{\Delta t}\) First, find the change in angular speed, Δω = ω_final - ω_initial: \(\Delta \omega = 2.55 - 7\) \(\Delta \omega = -4.45\; \mathrm{rad/s}\) Now, substitute the values into the angular acceleration formula: \(\alpha = \frac{-4.45}{18.00}\) Calculating the result, \(\alpha \approx -0.247 \; \mathrm{rad/s^2}\) So, the average angular acceleration of the take-up spool while the tape is being played is approximately -0.247 rad/s².

Key concepts

Angular Speed

Understanding angular speed is crucial when studying rotational motion. It is a measure of how fast an object rotates or spins in terms of angles. To define it simply, angular speed is the angle through which an object rotates in a specific unit of time, usually measured in radians per second (rad/s).

In the scenario of a tape recorder, maintaining a constant linear speed is essential for the correct playback of audio. The angular speed, however, changes depending on the radius of the take-up spool. When the spool is empty and has a smaller radius, the angular speed is higher to ensure that the linear speed of the tape being pulled remains constant. As the spool fills and its radius increases, less angular speed is needed to maintain the same linear speed. This inverse relationship between radius and angular speed is captured by the formula: \( \text{Angular Speed} (\text{ω}) = \frac{\text{Linear Speed} (v)}{\text{Radius} (r)} \).

Using this simple yet powerful relationship, we can calculate the changing angular speeds of any rotating object, provided we know its linear speed and varying radius at different points in time.

Constant Linear Speed

A constant linear speed implies that an object moves at a steady rate in a straight path, and its distance traveled over time remains uniform. This is a common scenario for conveyor belts, magnetic tape players, and vehicles cruising on a highway.

In our tape recorder example, the magnetic tape moves with a constant linear speed, which is the speed at which the tape unspools in the direction of its length. This steady speed ensures that the output sound from the tape is uniform and undistorted. To sustain this constant linear speed, despite the changing radius of the spool as the tape winds up, the recorder has to adjust the angular speed accordingly. As the radius increases, the angular speed decreases proportionally to keep the linear speed the same, showing a direct application of the aforementioned relationship.

Angular Acceleration

Angular acceleration refers to the rate at which the angular speed of an object changes over time. It is often denoted by the Greek letter alpha (α) and is measured in radians per second squared (rad/s²). Angular acceleration is an indicator of how quickly an object can increase or decrease its rotation speed.

Consider the tape recorder's take-up spool. As the tape winds onto the spool, the radius increases, causing a change in angular speed. The average angular acceleration, in this case, can be calculated by the change in angular speed divided by the time it takes for this change. It's important to note that since the spool slows down over time (a decrease in angular velocity), the angular acceleration is negative. Using the formula \( \text{Angular Acceleration} (α) = \frac{\text{Change in Angular Speed} (\text{Δω})}{\text{Change in Time} (\text{Δt})} \), we can quantify this deceleration. The exercise demonstrated how to calculate exactly that for a spool with decreasing angular speed, which is essential to understanding the dynamics of rotational systems.