Torque from moment of inertia

[physisis]

I have an assignment that asks to find a torque from given angular velocity, mass and moment of inertia.

For a turntable disk, mass is known as 0.04452199 kilograms

The Angular velocity should be 106.81 (rads/sec)

These values are from solidwork file (anyone familiar with Solidworks?)

Center of mass: ( meters )
X = -0.01474448
Y = -0.00659061
Z = 0.04376940

Principal axes of inertia and principal moments of inertia: ( kilograms *
square meters )
Taken at the center of mass.
Ix = (-0.70029829, 0.71385033, 0.00002079) Px = 0.00001872
Iy = (-0.71385033, -0.70029829, -0.00002111) Py = 0.00001875
Iz = (0.00000000, -0.00002962, 1.00000000) Pz = 0.00003073

Moments of inertia: ( kilograms * square meters )
Taken at the center of mass and aligned with the output coordinate system.
Lxx = 0.00001874 Lxy = -0.00000001 Lxz = 0.00000000
Lyx = -0.00000001 Lyy = 0.00001874 Lyz = 0.00000000
Lzx = 0.00000000 Lzy = 0.00000000 Lzz = 0.00003073

Moments of inertia: ( kilograms * square meters )
Taken at the output coordinate system.
Ixx = 0.00010596 Ixy = 0.00000431 Ixz = -0.00002873
Iyx = 0.00000431 Iyy = 0.00011371 Iyz = -0.00001284
Izx = -0.00002873 Izy = -0.00001284 Izz = 0.00004234As far as I know, equation for torque is angular acceleration X moment of inertia,
but in this case, I don't have Angular acceleration but velocity. and Radius is unknown.

Is it possible to get a value of Torque from these information? I am keep looking for way to approach this question but I am having a hard time with it.
Any help will be appreciated.

 

[RoyalCat]

Note that the turntable is spinning through an axis that is offset from its center of mass. That means that relative to its center of mass, the contact forces at the pivot produce a torque.

I think that the spin angular momentum vector goes through a precession because of this offset, but I'm not sure.

If I'm right in my approach, then the key to solving this question is: $$\vec \tau=\frac{d\vec L}{dt}$$ where in this case, $$\vec L$$ would be the spin angular momentum vector.

 

[kerimek]

Thank you for sharing the details of your assignment. To answer your question, yes, it is possible to calculate the torque from the given information. The equation for torque is indeed angular acceleration multiplied by moment of inertia, but in this case, we can use the angular velocity instead of acceleration.

To calculate the torque, we need to first find the radius of the turntable disk. This can be done by using the coordinates of the center of mass (X, Y, Z) and the principal moments of inertia (Px, Py, Pz). The radius can be calculated using the formula r = sqrt(X^2 + Y^2 + Z^2). In this case, the radius would be approximately 0.045 meters.

Next, we can use the given angular velocity (106.81 rads/sec) and the moment of inertia (Ixx, Ixy, Ixz, Iyx, Iyy, Iyz, Izx, Izy, Izz) to calculate the torque. The torque can be calculated using the formula T = I * ω, where I is the moment of inertia and ω is the angular velocity. Using this formula, we can calculate the torque for each axis (x, y, z) and then take the magnitude to get the total torque.

I hope this helps you with your assignment. Please let me know if you have any further questions.