Question

there is a small scratch on a disco record located 12.8 cm from the center. when played, the scratch makes the record skip 30 times each minute (i.e., the scratch passes under the record player needle 30 times per minute). what is the linear speed of the scratch as it turns? give your answer in units of cm/s

Answer 1

The linear speed of a scratch on a disco record located 12.8 cm from its center, making 30 revolutions per minute, is approximately 126.67 cm/s.

Step-by-step explanation:

The student is asking about the linear speed of a scratch on a record as it spins. Since the record makes the scratch pass under the needle 30 times each minute, we can calculate the linear speed using the formula ∑ = 2πr × (number of revolutions per second). We know that the scratch is 12.8 cm from the center, so the radius r is 12.8 cm. The record spins 30 times per minute, which is equivalent to 0.5 revolutions per second (30 rev/min × 1 min/60 sec).

The linear speed ∑ of the scratch can be calculated as:

∑ = 2 × π × 12.8 cm × 0.5 rev/sec = 40.32π cm/sec

After computing the value, we get:

∑ ≈ 126.67 cm/s

Therefore, the linear speed of the scratch as it turns on the disco record is approximately 126.67 cm/s.

Answer 2

The linear speed of the scratch as it turns is approximately 126.58 cm/s.

To find the linear speed of the scratch on the disco record as it turns, we need to calculate the distance the scratch travels in one revolution (i.e., one full turn) and then convert it to cm/s.

Step 1: Calculate the circumference of the circle traced by the scratch.

The circumference of a circle is given by the formula:

Circumference (C) = 2 * π * radius

In this case, the scratch is located 12.8 cm from the center, so the radius (r) of the circle traced by the scratch is 12.8 cm. Plug this value into the formula:

C = 2 * π * 12.8 cm

C = 25.6 π cm

Step 2: Calculate the distance traveled by the scratch in one revolution.

Since the scratch makes the record player needle skip 30 times each minute, it means the scratch passes under the needle 30 times in one minute. Therefore, in one full revolution (one turn), the scratch would pass under the needle 30 times.

So, the distance traveled by the scratch in one revolution is 30 times the circumference:

Distance in one revolution = 30 * C

Distance in one revolution = 30 * 25.6 π cm

Step 3: Calculate the linear speed in cm/s.

To find the linear speed, we need to know the time it takes for one revolution. Since there are 60 seconds in a minute (1 minute = 60 seconds), the scratch completes one revolution in 60 seconds.

Now, divide the distance traveled in one revolution by the time taken for one revolution to find the linear speed:

Linear Speed = (Distance in one revolution) / (Time for one revolution)

Linear Speed = (30 * 25.6 π cm) / 60 seconds

Now, let's calculate this:

Linear Speed ≈ (30 * 25.6 * 3.14159 cm) / 60 seconds

Linear Speed ≈ (2419.52 π cm) / 60 seconds

Linear Speed ≈ 40.325 π cm/s

Now, you can approximate this value to a decimal:

Linear Speed ≈ 40.325 * 3.14159 cm/s

Linear Speed ≈ 126.58 cm/s