Question

In a tape recorder, the magnetic tape moves at a constant linear speed of \(5.60 \mathrm{~cm} / \mathrm{s}\). 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 \(r_{1}=0.800 \mathrm{~cm} ?\) 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

Based on the given information, we were able to determine the following values: 1. The angular speed of the empty take-up spool is 7.00 rad/s. 2. The angular speed of the full take-up spool is 2.55 rad/s. 3. The average angular acceleration of the take-up spool while the tape is being played is -0.00248 rad/s^2. Using these values, provide a short answer discussing the motion of the tape during playback, the significance of the negative angular acceleration, and any possible implications for the playback quality of the tape.

Step-by-step solution

Part a: Angular speed of the empty take-up spool

To calculate the angular speed of the empty take-up spool, we can use the linear speed and the radius of the empty spool. The relationship between linear speed (v), angular speed (ω), and radius (r) is: $$v = \omega r$$ We can now plug in the given values and solve for the angular speed: $$5.60\,\text{cm/s} = \omega \cdot 0.800\,\text{cm}$$ Now, we solve for ω: $$\omega = \frac{5.60\,\text{cm/s}}{0.800\,\text{cm}} = 7.00\,\text{rad/s}$$ The angular speed of the empty take-up spool is \(\boxed{7.00\,\text{rad/s}}\).

Part b: Angular speed of the full take-up spool

We will use the same relationship to find the angular speed of the full take-up spool: $$v = \omega r$$ Plugging in the values for the linear speed and the radius of the full spool: $$5.60\,\text{cm/s} = \omega \cdot 2.20\,\text{cm}$$ Now, we solve for ω: $$\omega = \frac{5.60\,\text{cm/s}}{2.20\,\text{cm}} = 2.55\,\text{rad/s}$$ The angular speed of the full take-up spool is \(\boxed{2.55\,\text{rad/s}}\).

Part c: Average angular acceleration of the take-up spool

To find the average angular acceleration (α) of the take-up spool, we need to find the change in angular speed (Δω) and the time it takes for the tape to play (Δt). We can find the time it takes for the tape to play based on the length of the tape and its linear speed. Then, we can use the relationship between angular acceleration, change in angular speed, and time, given by: $$\alpha = \frac{\Delta \omega}{\Delta t}$$ First, let's find the time it takes for the tape to play: $$\Delta t = \frac{\text{length of tape}}{\text{linear speed}} = \frac{100.80\,\text{m}}{5.60\,\text{cm/s}} = \frac{10,080\,\text{cm}}{5.60\,\text{cm/s}} = 1800\,\text{s}$$ Now, we can find the change in angular speed by subtracting the initial angular speed (empty spool) from the final angular speed (full spool): $$\Delta \omega = \omega_\text{full} - \omega_\text{empty} = 2.55\,\text{rad/s} - 7.00\,\text{rad/s} = -4.45\,\text{rad/s}$$ Finally, we can find the average angular acceleration: $$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{-4.45\,\text{rad/s}}{1800\,\text{s}} = -0.00248\,\text{rad/s}^2$$ The average angular acceleration of the take-up spool while the tape is being played is \(\boxed{-0.00248\,\text{rad/s}^{2}}\).

Key concepts

Linear Speed

Linear speed is the distance an object travels per unit of time. In the context of a tape recorder, the magnetic tape moves at a certain linear speed to ensure the audio is played back at the correct rate. It is constant throughout the process to maintain an even playback speed. In the given exercise, the linear speed of the tape is mentioned as 5.60 cm/s, which means for every second, the tape travels 5.60 centimeters. Linear speed is important as it can directly relate to angular speed when considering rotating objects, such as spools of tape recorders.

Understanding linear speed helps in computing other elements in rotational motion, such as angular speed and acceleration, which are vital for the functioning of devices like tape recorders where rotational and linear movements are interconnected.

Angular Acceleration

Angular acceleration refers to the rate of change of angular velocity over time. In simpler terms, it measures how quickly something spins faster or slower. In our tape recorder scenario, as the tape winds onto the take-up spool, the spool's radius increases, causing a change in angular speed while maintaining a constant linear speed of the tape. The concept of angular acceleration becomes crucial to understand how the spool's speed changes over the playback time.

The exercise provided computes the average angular acceleration by considering the change in angular speed as the tape winds and the time taken for this change to occur. It reflects the gradual adjustment in the spinning rate of the spool to ensure it matches up with the constant linear speed of the tape being played.

Tape Recorder Physics

The physics behind a tape recorder is a fascinating illustration of how linear and rotational motions are intertwined. As a tape plays, its information—encoded on a magnetic strip—passes by a reading head at a consistent linear speed for accurate data retrieval. This is crucial for maintaining the pitch and tempo of the audio. However, because the tape is wound around a spool, as the amount of tape on the spool changes, the spool's radius changes as well.

Therefore, the angular speed of the spool must adjust correspondingly, as a constant linear speed coupled with varying spool radius results in varying angular speeds. This dynamic showcases the intricate balance necessary in the mechanical design of a tape recorder to maintain consistency in audio playback.

Relationship Between Linear and Angular Velocities

Understanding the relationship between linear and angular velocities is critical to solving problems related to rotational motion. Linear velocity pertains to how fast a point is moving along a path, whereas angular velocity denotes how fast the angle is changing as an object rotates. The two are interconnected by the radius of the rotation: linear speed (v) = angular speed (ω) * radius (r).

In the context of our tape recorder example, the magnetic tape has a constant linear speed, and the radius of the take-up spool changes as the tape is played. This changes the angular speed according to the relationship v = ωr. As the radius increases, the angular speed must decrease to maintain that constant linear speed of the tape. This relationship is crucial in calculating angular speed adjustments and ensuring the tape recorder functions correctly.