Angular Acceleration of a Wheel on a Turntable

In summary: So the direction would be ##\vec{\alpha} = \omega_1\cos(\omega_2 t)\hat{i} + \omega_1\sin(\omega_2 t)\hat{j}##
  • #1
bsvh
6
0

Homework Statement


[/B]
The axle of a wheel is mounted on supports that rest on a rotating turntable. the wheel has angular velocity ##\omega_1 = 44.0\; \frac{\textrm{rad}}{\textrm{s}}## about its axle, and the turntable has angular velocity ##\omega_2 = 35.0\; \frac{\textrm{rad}}{\textrm{s}}## about a vertical axis. What is the magnitude and direction of the angular acceleration of the wheel at the instant shown? Take the ##z## axis vertically upward and the direction of the axle at the moment shown to be the ##x## axis pointing to the right.

2x3l1z8.png


Homework Equations



##\vec{\alpha} = \frac{\textrm{d}\vec{\omega}}{\textrm{d}t}##

The Attempt at a Solution



This problem stumped me a bit, but I think I got it now and just want to make sure. ##\vec{\omega}_2##, which is the angular velocity of the turntable, is constant because using the right hand rule it is always pointing up. ##\vec{\omega}_1## varies because the turntable causes the direction of it to change, although its magnitude is the same. Since ##\vec{\alpha} = \frac{\textrm{d}\vec{\omega}}{\textrm{d}t}##, and ##\vec{\omega}_2## is constant, the acceleration is then ##\frac{\textrm{d}\vec{\omega}_1}{\textrm{d}t}##.

The direction of ##\vec{\omega}_1## follows the circular path of ##\vec{\omega}_2##, so it can be parameterized by:

##\vec{\omega}_1 = \omega_1 \cos (\omega_2 t)\hat{i} + \omega_1 \sin(\omega_2 t)\hat{j}##

If we take ##t = 0## to be shown by the picture above, then at ##t=0## ##\vec{\omega}_1## is pointing to the left, so it has to be negative. Therefore, the parametrization should be:

##\vec{\omega}_1 = -\omega_1 \cos (\omega_2 t)\hat{i} - \omega_1 \sin(\omega_2 t)\hat{j}##

To get the acceleration, I take the derivative with respect ot time and find that:

##\vec{\alpha} = \omega_1\omega_2\sin(\omega_2 t)\hat{i} - \omega_1\omega_2\sin(\omega_2 t)\hat{j}##

I then simply have to find the magnitude and direction of this to get the answers. Was my process correct?
 
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  • #2
Check the j term in your final equation.
 
  • #3
haruspex said:
Check the j term in your final equation.

Whoops! should be cosine.
 

1. What is angular acceleration?

Angular acceleration is the rate of change of angular velocity with respect to time. In simpler terms, it is the measure of how quickly an object's rotational speed is changing.

2. How is angular acceleration calculated?

Angular acceleration is calculated by dividing the change in angular velocity by the change in time. The formula for angular acceleration is: α = (ω2 - ω1) / (t2 - t1), where α is the angular acceleration, ω is the angular velocity, and t is the time.

3. What factors affect the angular acceleration of a wheel on a turntable?

The angular acceleration of a wheel on a turntable is affected by the radius of the wheel, the mass of the wheel, the moment of inertia of the wheel, and the applied torque. The larger the radius and mass of the wheel, the smaller the angular acceleration will be. The larger the moment of inertia, the greater the angular acceleration will be. And the greater the applied torque, the greater the angular acceleration will be.

4. How does angular acceleration differ from linear acceleration?

Angular acceleration refers to the change in rotational speed, while linear acceleration refers to the change in linear speed. Angular acceleration is measured in radians per second squared, while linear acceleration is measured in meters per second squared.

5. How does the angular acceleration of a wheel on a turntable affect its motion?

The angular acceleration of a wheel on a turntable determines how quickly the wheel will rotate. A greater angular acceleration will result in a faster rotation, while a smaller angular acceleration will result in a slower rotation. This also affects the overall motion of the turntable, as the rotation of the wheel will cause the turntable to move in a circular motion.

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