Gyroscope- Angular Speed, Rotational Kinematics, Precession

In summary, to bring a gyroscope up to speed, it requires 7.45×10^4 watts of power. The torque necessary to cause it to precess is 1422.84 Newtons.
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jcwhitts
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Gyroscope-- Angular Speed, Rotational Kinematics, Precession

Homework Statement



A Gyro Stabilizer. The stabilizing gyroscope of a ship is a solid disk with a mass of 6.50×10^4 kg; its radius is 2.30m, and it rotates about a vertical axis with an angular speed of 500 rev/min.

1) How much time is required to bring it up to speed, starting from rest, with a constant power input of 7.45×10^4 W?

2)Find the torque needed to cause the axis to precess in a vertical fore-and-aft plane at an angular rate of 1.00 degree/sec.



Homework Equations


I may be incorrect as I apparently keep coming up with the wrong answer. However, the moment of inertia for a solid cylinder I=1/2mr^2. P=power, I=moment of inertia, w=angular speed, a=angular acceleration, T=torque.

P=Tw. Ia=T. W=at.

For the second question, I'm a lot more lost. I'm guessing it's Torque/angular momentum but I continue to get incorrect answers.

The Attempt at a Solution



Ok, perhaps it's an issue with me actually understanding what is given because I would assume you would need a length from a pivot in order to work with a gyroscope. Since there is no such info in the problem, I assume that the disk is rotating directly on the pivot point.

I started with finding I. I=1/2(6.5*10^4)(2.3^2). I get 171925. Next, I use the Power equation to find the torque, using the given power and target angular speed. 7.45*10^4=T(500(rev/min). I convert 500 Rev/min to rads per second using 500*(2pi/60) to get 52.3599.

Then, solving for T, (7.45*10^4)/52.3599 to get 1422.84.

T=Ia. Using 1422.84=171925a, we find alpha to be .008276. Here is where I think there is something I may not be accounting for with torque due to the weight of the disc, but I assumed that since it is acting at the center of mass there is no torque. I tried the solution with torque being equal to mg + 1422.84, and the answer is still wrong.

Anyway, now that we have a, and given that the gyro starts from rest, w=at. So, 52.3599/.008276 = t, which =6327.

This answer is incorrect. Am I making incorrect assumptions somewhere as to using equations that wouldn't apply in this situation or something I am missing completely? Any help is much appreciated.

As for the precession part, that can wait until after I've figured out the first part.
 
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  • #2


Any ideas?
 
  • #3


Figured it out. My approach went off on a tangent because the angular velocity is not constant. Instead, you use the energy approach K=1/2Iw^2 and know that the work is equal to the change in kinetic energy, and Power is work over time. So, Power * change in time =W.

1/2Iw^2=P*t, divide by the given power.
 

1. What is a gyroscope?

A gyroscope is a device used to measure or maintain orientation and angular velocity. It consists of a spinning rotor that is mounted on a set of gimbals, allowing it to rotate freely in any direction.

2. What is angular speed?

Angular speed, also known as rotational speed, is the rate at which a rotating object turns around a central axis. It is measured in units of radians per second or revolutions per minute.

3. How does a gyroscope measure angular velocity?

A gyroscope uses the principle of conservation of angular momentum to measure angular velocity. As the rotor spins, it resists any changes in its orientation, allowing it to accurately detect changes in angular velocity.

4. What is rotational kinematics?

Rotational kinematics is the study of the motion of objects that are rotating around a fixed axis. It involves concepts such as angular displacement, angular velocity, and angular acceleration.

5. What is precession in a gyroscope?

Precession in a gyroscope refers to the change in orientation of the spinning rotor due to an external torque. This phenomenon is used in gyroscopic stabilization systems, such as those found in airplanes and ships.

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