Compact Disc. A compact disc (CD) stores music in a coded patt...

Question

Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits 10^-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. (c) What is the average angular acceleration of a maximumduration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.

Accepted Answer

Hey everyone. So today we're dealing with the problem about angular acceleration. So we're being told that we have a sewing machine that has a speed of stitches per minute a stitches, two millimeters, giving a constant linear sewing speed of 9200 millimeters per minute. Now the sewing machine is performing embroidery on rotating fabric. So the pattern of the embroidery is a spiral circle whose radius increases outwards, and the inner radius of the pattern is 0.8 centimeters while the outer radius is 10 centimeters. If the embroidery is completed in 25 minutes, determined the we're being asked to determine the average angular acceleration of the fabric for said 25 minute period. We're also being asked to take the direction of angular velocity as the positive direction. So let's write down what we know first, and let's also convert them into our standard units. We know that we have our time. Our change in time Change in time is 25 minutes. We'd like this in seconds. And our base units who recall that we have 60 seconds for every one minute. So this gives us 15:00. We also have our velocity, it's constant, It's our constant linear sewing speed and that is per minute. We also want to convert this to our basic units of meters per second. So we can convert using a few conversion factors first recalled at one m Contains 1000. And again, We know that one minute Is equal to 60 seconds. So simplifying everything. We get that the linear sewing speed, the constant linear sewing speed or the constant linear velocity,  aN:aN:000NaN 0. m/s. We also know the radius of the inner pattern R. And I'll put in er It's 0. cm But we want this in meters so we can multiply by one m that contains 100 cm. That's our conversion factor. So we get 0. meters and the outer radius can be done in a similar fashion. We have 10 cm Recall that one m is nothing but 100 cm. Our center musical cancel out and we will be left with 0. m. So these are our current Parameters with this. We need to go ahead and look closely and use the formula for angular acceleration. Now the formula for angular acceleration. And let me scroll down so I just have a little bit more space. But the formula for angular acceleration averaging angular acceleration is the difference in the angular velocities. The final velocity minus the initial angular velocity over the change in time, which means we need to find the angular velocity at the end, at the outermost and at the innermost which is the initial point. So let's use that in red to find the initial angular velocity. It's simply the velocity divided by our inner which is simply 0. m/s over 0.008 0.008 m which gives us Once we convert one point or sorry, 19.1 radiance per second. Similarly we can do the same thing for the final angular velocity which is simply the velocity over the outer radius. So that will be 0.153 m/s over 0.1 m. So that'll equate to 1. radiance per second. So with that in hand we can go ahead and plug that back into our equation. Our handy dandy equation over here. So our final is 1.53 radiance per second minus 19. radiance per second Over the change in time which we determined to be 15:00. So solving this gives us a final answer Of negative 0. radiance per second squared. And that's our angular acceleration or average angular acceleration Over that 25 minute period. Now this is important because the negative acceleration means that the angular velocity is decreasing and the angular and the acceleration is directed against the direction of the velocity. So it's slowing down eventually. But with this answer choice, this lines up with answer choice, d -0.0117 radiance per second squared. I hope this helps. And I look forward to seeing you all in the next one

Key Concepts

Angular Acceleration

Relationship Between Linear and Angular Quantities

Kinematics of Rotational Motion

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