Torque <-> rotational velocity?

In summary, the problem the OP is having is trying to calculate the rotational velocity of a rigid object given a force, which can be done by doing the cross product of the force vector and the offset vector, the offset being the difference between the point on which the force is applied and the centre of mass for the object. However, the rotational velocity must be calculated in global Euler angles, which is a bit of a problem because the answer to this question is hard to find. However, by applying the torque directly, the problem is solved.
  • #1
Hnefi
2
0
Hello. I'm having a bit of a problem. I need to calculate a change in rotational velocity on a rigid object given a force (actually an impulse, but nevermind) acting on that object.

I can calculate the torque without problem, by doing the cross product of the force vector and the offset vector, the offset being the difference between the point on which the force is applied and the centre of mass for the object. Fine. But I need to translate that into rotational velocity, or rather change in rotational velocity.

The object has a known mass and the rotation must be calculated in global Euler angles (as in it doesn't matter how the object is rotated or what its current angular velocity is). How do I do this? I tried projecting the force vector onto the cross product between the offset vector and the torque, but that didn't work. I've been searching the 'net, but for some reason the answer to this question is pretty darn hard to find.

Thanks in advance for any help.
 
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  • #2
Hnefi said:
Hello. I'm having a bit of a problem. I need to calculate a change in rotational velocity on a rigid object given a force (actually an impulse, but nevermind) acting on that object.

I can calculate the torque without problem, by doing the cross product of the force vector and the offset vector, the offset being the difference between the point on which the force is applied and the centre of mass for the object. Fine. But I need to translate that into rotational velocity, or rather change in rotational velocity.

The object has a known mass and the rotation must be calculated in global Euler angles (as in it doesn't matter how the object is rotated or what its current angular velocity is). How do I do this? I tried projecting the force vector onto the cross product between the offset vector and the torque, but that didn't work. I've been searching the 'net, but for some reason the answer to this question is pretty darn hard to find.

Thanks in advance for any help.
Hi Hnefi and welcome to PF,

Am I missing something here, or isn't this just a simple application of,

[tex]\sum_i \boldmath{F}_i\times\boldmath{r}_i = \mathbb{I}_G\frac{d}{dt}\boldmath{\omega}[/tex]
 
  • #3
Ack, you're right. I overengineered the problem. Simply applying the torque directly did the trick. Thanks for your help.
 
  • #4
Hootenanny said:
Hi Hnefi and welcome to PF,

Am I missing something here, or isn't this just a simple application of,

[tex]\sum_i \boldmath{F}_i\times\boldmath{r}_i = \mathbb{I}_G\frac{d}{dt}\boldmath{\omega}[/tex]

Not quite that simple. In inertial coordinates, the correct expression is

[tex]\sum_i \mathbf{F}_i\times\mathbf{r}_i
= \frac{d\mathbf L_I}{dt}
= \frac{d}{dt}\left({\mathbb{I}}_I\,\mathbf{\omega}_I\right)[/tex]

where the subscript I on the angular momentum [itex]\mathbf L[/itex], inertia tensor [itex]\mathbb{I}[/itex] and angular velocity [itex]\boldmath{\omega}[/itex] indicate that the quantities in question are to be expressed in terms of a non-rotating (i.e. inertial) frame. The problem is that the inertia tensor for a rigid body as observed from an inertial frame is not constant. The inertia tensor for a rigid body is constant in a body-fixed frame. It is much more convenient to do the calculations in the body-fixed (i.e., rotating) frame. However, this means that one must introduce a fictitious torque.

The transport theorem relates the time derivative of some vector quantity [itex]\mathbf q[/itex] from the perspective of an inertial observer and a body-fixed observer:

[tex]\frac{d \mathbf q}{dt_I} = \frac{d \mathbf q}{dt_B} + \mathbf{\omega} \times \mathbf q[/tex]

With [itex]\mathbf q = \mathbf L[/itex], this becomes

[tex]\frac{d \mathbf L}{dt_I} = \mathbb{I} \frac{d \mathbf{\omega}}{dt_B} + \mathbf{\omega} \times (\mathbb{I} \,\mathbf{\omega})[/tex]

Combining with the first equation,

[tex]\frac{d \mathbf \omega_B}{dt}
= \mathbb{I}_B^{\;-1}\left(\sum_i \mathbf{F}_{i_B}\times \mathbf{r}_{i_B} - \mathbf{\omega}_B \times (\mathbb{I}_B\, \mathbf{\omega}_B)\right)[/tex]

Here the subscripts B denote that the quantities in question, including external forces, are to be expressed in body-fixed coordinates.
 
Last edited:
  • #5
I stand corrected DH. In all honesty I thought the OP mentioned body angles, but I see that I was mistaken.

I doff my cap to you sir, nice post!
 

What is torque and rotational velocity?

Torque is a measure of the force that causes an object to rotate around an axis. It is usually represented by the symbol "τ". Rotational velocity, also known as angular velocity, is the rate at which an object rotates around an axis. It is typically represented by the symbol "ω".

How are torque and rotational velocity related?

Torque and rotational velocity are directly proportional to each other. This means that an increase in torque will result in an increase in rotational velocity, and vice versa. The exact relationship between the two is described by the equation τ = Iω, where "I" represents the moment of inertia of the object.

What is the SI unit for torque and rotational velocity?

The SI unit for torque is Newton-meter (Nm), while the SI unit for rotational velocity is radians per second (rad/s). However, other units such as foot-pound (ft-lb) and revolutions per minute (RPM) are also commonly used to measure torque and rotational velocity, respectively.

How does torque and rotational velocity affect the motion of an object?

Torque and rotational velocity determine the angular acceleration of an object. If the net torque on an object is zero, it will maintain a constant rotational velocity. On the other hand, if there is a non-zero net torque, the object will experience an angular acceleration, which will result in a change in rotational velocity.

What are some real-life applications of torque and rotational velocity?

Torque and rotational velocity are essential in many everyday objects and machines, such as bicycles, cars, and household appliances. They are also crucial in understanding the motion of celestial bodies, such as planets and stars. In industries, torque and rotational velocity are used in the design and operation of engines, turbines, and other rotating machinery.

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