Rotation of a disk - find angular acceleration

In summary, the problem involves a disk rotating with constant angular acceleration, starting from rest and reaching a final angular speed of 33 rev/s after completing 69 revolutions. The summary includes the calculations for (a) the angular acceleration, (b) the time required to complete the 69 revolutions, (c) the time required to reach the 12 rev/s angular speed, and (d) the number of revolutions from rest until the disk reaches 12 rev/s angular speed. The calculations are correct and the problem has been solved accurately.
  • #1
blue5t1053
23
1
Problem:
A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at 12 rev/s; 69 revolutions later, its angular speed is 33 rev/s. Calculate (a) the angular acceleration, (b) the time required to complete the 69 revolutions, (c) the time required to reach the 12 rev/s angular speed, and (d) the number of revolutions from rest until the time the disk reaches 12 rev/s angular speed.

My work:
[tex]part (a):
\omega ^{2} = \omega ^{2}_{0} + 2 \alpha (\vartheta - \vartheta _{0}) \Rightarrow \frac{2 * \pi * (33^{2} rev/s - 12^{2} rev/s)}{2 * 69 sec} = 43.03 rad/s^{2}[/tex]

[tex]part (b):
\vartheta =\vartheta \frac{1}{2} (\omega _{0} + \omega)* t \Rightarrow \frac{2 * 69 sec}{12 rev/s + 33 rev/s} = 3.07 sec[/tex]

[tex]part (c):
\omega ^{2} = \omega ^{2}_{0} + 2 \alpha (\vartheta - \vartheta _{0}) \Rightarrow \frac{(33^{2} rev/s - 12^{2} rev/s)}{2 * 69 sec} = 6.85 rev/s^{2};[/tex]
[tex]\omega = \omega _{0} + \alpha * t \Rightarrow \frac{12 rev/s}{6.85 rev/s^{2}} = 1.75 sec[/tex]

[tex]part (d):
\omega ^{2} = \omega ^{2}_{0} + 2 \alpha (\vartheta - \vartheta _{0}) \Rightarrow \frac{(12^{2} rev/s - 0^{2} rev/s)}{2 * 6.85 rev/s^{2}} = 10.5 rev[/tex]

I think I did it correctly, but I would appreciate if I could have my work checked since it's the first time I've done angular acceleration. Thank you.
 
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  • #2
Looks Good.
 
  • #3


Your work looks correct to me! You used the correct equations and units and your calculations are consistent. Great job!
 

1. How is angular acceleration defined for a rotating disk?

Angular acceleration is defined as the rate of change of angular velocity with respect to time. It is measured in radians per second squared and is denoted by the symbol α.

2. What factors affect the angular acceleration of a disk?

The angular acceleration of a disk is affected by the net torque applied to it, the moment of inertia of the disk, and the distance from the axis of rotation at which the torque is applied. It is also influenced by any external forces acting on the disk.

3. Can the direction of angular acceleration change for a rotating disk?

Yes, the direction of angular acceleration can change for a rotating disk. This usually occurs when the direction of the net torque changes, causing the angular velocity to either increase or decrease in magnitude.

4. How is angular acceleration calculated for a rotating disk?

The formula for calculating angular acceleration is α = Δω/Δt, where α is the angular acceleration, Δω is the change in angular velocity, and Δt is the change in time.

5. How does the angular acceleration of a disk relate to its linear acceleration?

The angular acceleration of a disk is directly related to its linear acceleration through the equation α = a/r, where α is the angular acceleration, a is the linear acceleration, and r is the radius of the disk. This relationship is known as the tangential acceleration equation.

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