Question

In the professional videotape recording system known as quadriplex, four tape heads are mounted on the circumference of a drum of radius \(2.5 \mathrm{cm}\) that spins at \(1500 \mathrm{rad} / \mathrm{s} .\) (a) At what speed are the tape heads moving? (b) Why are moving tape heads used instead of stationary ones, as in audiotape recorders? [Hint: How fast would the tape have to move if the heads were stationary?]

Short answer

Answer: The linear velocity of the tape heads is 3750 cm/s. Moving tape heads are preferred over stationary ones because they allow for lower tape speed while still achieving the desired recording quality, reducing wear and tear on the tape and mechanical stress on the system.

Step-by-step solution

Identify the given information

We are given: - The radius of the drum, \(r = 2.5\) cm - The angular velocity, \(\omega = 1500\) rad/s We need to find the linear velocity of the tape heads, denoted as \(v\).

Use the formula for linear velocity

To find the linear velocity, we will use the following formula: \(v = r × \omega\) Where, - \(v\) is the linear velocity of the tape heads - \(r\) is the radius of the drum - \(\omega\) is the angular velocity of the spinning drum

Plug in the given values and calculate the linear velocity

Plugging in the given values for the radius and angular velocity, we get: \(v = (2.5 \ \text{cm}) × (1500 \ \text{rad/s})\) \(v = 3750 \ \text{cm/s}\) Therefore, the linear velocity of the tape heads is 3750 cm/s.

Reason why moving tape heads are used instead of stationary ones

Moving tape heads are used in professional videotape recording systems because they allow for lower tape speed while still achieving the desired recording quality. If stationary tape heads were used, the tape would have to move at a much faster speed (in this case, 3750 cm/s) to achieve the same recording quality. This would result in increased wear and tear on the tape and additional mechanical stress on the entire system. In conclusion, the tape heads in the quadriplex recording system move at a linear velocity of 3750 cm/s. This is advantageous compared to stationary tape heads, as it allows for lower tape speeds while still achieving high-quality recording.