# Find torque in the x direction given moment of inertia tensor for plate.

• AbigailM
In summary, the plate's angular momentum is changing due to the rotation of the perpendicular component of its moment of inertia. The rate of change of the angular momentum is given by the equation \tau=\sigma l^{4} \omega^{2} \boldsymbol{\hat{z}}.
AbigailM
This is for prelim study. Just wondering if this solution is correct.

Problem

A thin homogeneous plate lies in the x-y plane. Its moment of inertia tensor in the x,y,z basis is given by

$\textbf{I}=σl^{4}\begin{pmatrix} 2 & -2 &0 \\ -1 & 2 & 0 \\ 0 & 0 & 4\end{pmatrix}$

If the plate is rotated about the $\hat{x}$-axis with constant angular velocity ω, what torque must be applied to the plate to keep the roation axis pointing in the x direction?

The attempt at a solution

We are given $\textbf{I}$ and we know that $\textbf{ω}=(ω,0,0)$.

$\textbf{L}=\textbf{Iω}=σl^{4}\begin{pmatrix} 2 & -2 &0 \\ -1 & 2 & 0 \\ 0 & 0 & 4\end{pmatrix}\begin{pmatrix}ω \\ 0\\ 0\end{pmatrix}=σl^{4}ω(2,-1,0)$

$L_{x}=2σl^{4}ω$

$\Gamma_{x}=\frac{dL_{x}}{dt}=2\sigma l^{4}\dot{\omega}$

Thanks for the help.

Not quite.

As you can see, the angular momentum vector L is not parallel to the rotation vector ω, which means that a torque must be applied to the body in order to keep the rotation axis fixed along the x-axis with the angular momentum vector rotating around it.

If you imagine a diagram of L as a function of time you can see that the tip of the vector must be going in a circle around the x-axis with the angular speed of ω. In order for it to do so, the torque must be orthogonal to both L (to keep the magnitude of L constant) and ω (to keep the rotational speed around the x-axis constant), meaning that the torque must be parallel to ωxL and its magnitude proportional to ω.

Hmm. Shouldn't the moment of inertia tensor be symmetric?

Anyway, your calculation gives the angular momentum at the instant when the plate is oriented in the xy plane. At that instant, L has an x component and a y component. So, L lies within the plane of the plate. As the plate rotates, the x-component of L remains constant, but the component that is perpendicular to the x-axis will rotate with the plate so that L always remains oriented in the plane of the plate. So, from the point of view of our fixed inertial coordinate system, the angular momentum of the plate is changing with time due to the rotation of the component of L that is perpendicular to the rotation axis.

Can you find the rate of change of L due to the rotation of the component of L that is perpendicular to the rotation axis?

[I see now that Filip already gave a similar answer. Sorry]

Thank you both for the help.

After using the advice given above:

$\boldsymbol{\omega}=(\omega,0,0)$

$\boldsymbol{L}=\sigma l^{4} \omega(2,-1,0)$

$\boldsymbol{\tau}=\boldsymbol{\omega}\hspace{1 mm}\times \hspace{1 mm}\boldsymbol{L}$

$\boldsymbol{\tau}=\sigma l^{4} \omega^{2} \boldsymbol{\hat{z}}$

Just wanted to make sure all looks well. Thanks again for all the help.

## What is torque?

Torque is a measure of the force that causes an object to rotate around an axis. It is calculated by multiplying the force applied to the object by the distance from the axis of rotation.

## What is the x direction in relation to torque?

The x direction is a specific direction or axis in which torque is being calculated. It is typically perpendicular to the plane of rotation and is defined by the right-hand rule.

## What is a moment of inertia tensor?

A moment of inertia tensor is a mathematical representation of an object's resistance to changes in rotation. It takes into account the mass distribution and shape of the object to determine how much torque is required to rotate it.

## How do you find torque in the x direction?

To find torque in the x direction, you would need to use the moment of inertia tensor for the object and the force applied to it. The formula for torque in the x direction is T = Ixω, where Ix is the moment of inertia in the x direction and ω is the angular velocity.

## Can torque in the x direction be negative?

Yes, torque in the x direction can be negative. This indicates that the object is rotating in the opposite direction of the applied force. A positive torque indicates rotation in the same direction as the applied force.

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