Direction of an infinitesimal rotation?

In summary, the conversation discusses the torque on a magnetic dipole and the work done during rotation. It is mentioned that the direction of the infinitesimal rotation depends on the direction of the torque, which can be determined using the right hand rule. The question is posed on how to determine the direction of the infinitesimal rotation in the general case.
  • #1
etotheipi
I had a question from the magnetic dipole thread that was posted earlier today, but it's a bit more mundane. The torque on a magnetic dipole, using a right handed cross product is ##\vec{\tau} = \vec{\mu} \times \vec{B}##. The work done during a rotation is $$W = \int \vec{F} \cdot d\vec{r} = \int \vec{F} \cdot d\vec{\theta} \times \vec{r} = \int \vec{r} \times \vec{F} \cdot d\vec{\theta} = \int \vec{\tau} \cdot d\vec{\theta}$$Now the result of this depends on which direction we take the vector ##d\vec{\theta}## to be in, and there is only one correct result for work! We know it has to be parallel to the axis of rotation, but get the right answer ##d\vec{\theta}## also has to be in the opposite direction to ##\vec{\tau}##, i.e. if ##\vec{\tau} = \tau \hat{z}## then ##d\vec{\theta} = -d\theta \hat{z}##.

My question is, in the general case, how is the direction of the infinitesimal rotation determined, out of the two possible choices? I apologise if I am missing something obvious.
 
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  • #2
Dont you still follow the right hand rule? And the fact that the integral goes from one value of theta to and another value of theta.

When no limits are shown we assume the integral is positive and that the thetas range from low to high Ie counterclockwise.

Calling @fresh_42
 
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  • #3
jedishrfu said:
Dont you still follow the right hand rule? And the fact that the integral goes from one value of theta to and another value of theta.

It's difficult for me to tell :wink:. If we take ##d\vec{\theta}## in the opposite direction to the torque then we get $$W = \int_{\theta_1}^{\theta_2} -\mu B \sin{\theta} d\theta$$and I suppose this choice of direction for ##d\vec{\theta}## would correspond to measuring the angle ##\theta## from ##\vec{B}## to ##\vec{\mu}##, as opposed to from ##\vec{\mu}## to ##\vec{B}##.
 
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Related to Direction of an infinitesimal rotation?

1. What is an infinitesimal rotation?

An infinitesimal rotation is a mathematical concept that describes a very small change in the orientation of an object or system. It is often used in physics and engineering to model the movement of rigid bodies.

2. How is the direction of an infinitesimal rotation determined?

The direction of an infinitesimal rotation is determined by the axis of rotation, which is a line that passes through the center of rotation and remains fixed during the rotation. The direction of the rotation is perpendicular to this axis.

3. What is the difference between an infinitesimal rotation and a finite rotation?

An infinitesimal rotation is a very small change in the orientation of an object, while a finite rotation is a larger change. Infinitesimal rotations are often used to approximate finite rotations, especially when dealing with complex systems.

4. How is the direction of an infinitesimal rotation represented mathematically?

The direction of an infinitesimal rotation can be represented using a vector, with the magnitude of the vector representing the angle of rotation and the direction of the vector representing the axis of rotation.

5. What are some real-world applications of infinitesimal rotations?

Infinitesimal rotations are used in a wide range of fields, including robotics, computer graphics, and aerospace engineering. They are also used in physics to study the behavior of particles and in mathematics to solve complex differential equations.

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