Three disks are spinning independently on the same axle without friction. Their respective rotational inertias and angular speeds are I,ω𝐼,𝜔 (clockwise); 2I,3ω2𝐼,3𝜔 (counterclockwise); and 4I,ω/24𝐼,𝜔/2 (clockwise). The disks then slide together and stick together, forming one piece with a single angular velocity. What will be the direction and the rate of rotation ωnet𝜔net of the single piece?Express your answer in terms of one or both of the variables I𝐼 and ω𝜔 and appropriate constants. Use a minus sign for clockwise rotation.View Available Hint(s)for Part HActivate to select the appropriates template from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeActivate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeωnet𝜔net =
Question
Three disks are spinning independently on the same axle without friction. Their respective rotational inertias and angular speeds are I,ω𝐼,𝜔 (clockwise); 2I,3ω2𝐼,3𝜔 (counterclockwise); and 4I,ω/24𝐼,𝜔/2 (clockwise). The disks then slide together and stick together, forming one piece with a single angular velocity. What will be the direction and the rate of rotation ωnet𝜔net of the single piece?Express your answer in terms of one or both of the variables I𝐼 and ω𝜔 and appropriate constants. Use a minus sign for clockwise rotation.View Available Hint(s)for Part HActivate to select the appropriates template from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeActivate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeωnet𝜔net =
Solution 1
The conservation of angular momentum states that the total angular momentum before the disks stick together is equal to the total angular momentum after they stick together.
Before they stick together, the total angular momentum is the sum of the angular momenta of the three disks. The angular momentum of a rotating object is given by the product of its rotational inertia and its angular speed.
For the first disk, the angular momentum is Iω (clockwise, so we'll consider it as negative). For the second disk, the angular momentum is 2I3ω (counterclockwise, so it's positive). For the third disk, the angular momentum is 4I*(ω/2) (clockwise, so it's negative).
So, the total angular momentum before they stick together is -Iω + 6Iω - 2Iω = 3Iω.
After they stick together, the total rotational inertia is the sum of the rotational inertias of the three disks, which is I + 2I + 4I = 7I.
Let's denote the final angular velocity as ωnet. According to the conservation of angular momentum, we have:
7I * ωnet = 3Iω.
Solving for ωnet, we get:
ωnet = 3Iω / 7I = 3ω / 7.
Since the result is positive, the final rotation is counterclockwise.
Solution 2
The conservation of angular momentum states that the total angular momentum before the disks stick together is equal to the total angular momentum after they stick together.
Before they stick together, the total angular momentum is the sum of the angular momenta of the three disks. The angular momentum of a rotating object is given by the product of its rotational inertia and its angular speed.
For the first disk, the angular momentum is Iω (clockwise, so we'll consider it as negative). For the second disk, the angular momentum is 2I(-3ω) (counterclockwise, so it's positive). For the third disk, the angular momentum is 4I*ω (clockwise, so it's negative).
So, the total angular momentum before they stick together is -Iω + 2I3ω - 4Iω/2 = 6Iω - 2Iω = 4Iω.
After they stick together, the total angular momentum is the product of the total rotational inertia and the angular speed of the single piece. The total rotational inertia is the sum of the rotational inertias of the three disks, which is I + 2I + 4I = 7I.
So, the total angular momentum after they stick together is 7I*ωnet.
Setting these two equal gives 4Iω = 7I*ωnet.
Solving for ωnet gives ωnet = 4ω/7.
Since the result is positive, the direction of rotation is counterclockwise.
So, the single piece will rotate at a rate of 4ω/7 in the counterclockwise direction.
Similar Questions
A uniform disk with mass 41.0 kgkg and radius 0.220 mm is pivoted at its center about a horizontal, frictionless axle that is stationary. The disk is initially at rest, and then a constant force 29.5 NN is applied tangent to the rim of the disk.Part AWhat is the magnitude v𝑣 of the tangential velocity of a point on the rim of the disk after the disk has turned through 0.190 revolution?Express your answer with the appropriate units.Activate to select the appropriates template from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeActivate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeActivate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value typev𝑣 =ratnothing
A wooden disk with a moment of inertia of 5 kg m2 rotates with an angular speed of 50 rad/s. A metal cylinder with rotational inertia 20 kg m2 is dropped from rest onto the wooden cylinder so that their centers coincide.(a) When the cylinders no longer slip against each other, what is the final angular speed of the rotating system?
A point on the edge of a disk rotates around the center of the disk with an initial angular velocity of 3rad/s clockwise. The graph shows the point’s angular acceleration as a function of time. The positive direction is considered to be counterclockwise. All frictional forces are considered to be negligible.QuestionHow can the graph be used to determine the angular speed of the disk after 2s? Justify your selection.ResponsesDetermine the area bound by the line and the horizontal axis from 0s to 2s , because this area represents the final angular velocity of the point on disk.Determine the area bound by the line and the horizontal axis from 0 seconds to 2 seconds , because this area represents the final angular velocity of the point on disk.Determine the area bound by the line and the horizontal axis from 0s to 2s , because this area represents the change in the angular velocity of the point on disk.Determine the area bound by the line and the horizontal axis from 0 seconds to 2 seconds , because this area represents the change in the angular velocity of the point on disk.Determine the vertical-axis intercept, because this value represents the final angular velocity of the point on disk.Determine the vertical-axis intercept, because this value represents the final angular velocity of the point on disk.Determine the slope of the line, because this value represents the change in the angular velocity of the point on disk.
At a certain instant, a compact disc is rotating at 210 rpm. What is its angular speed in rad/s?
When finding the angular speed, you have to multiply the revolutions per second by .Group of answer choices𝜋 radians180∘180∘𝜋 radians2𝜋 radians1 revolution1 revolution2𝜋 radians PreviousNext
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.