Rotational Mechanics:Small Problems

[sganesh88]

I'd like to simply lay down a couple of ideas first that might help. Thinking of a particle translating in a circular trajectory with constant speed, the centripetal force is always perpendicular to the trajectory (or velocity vector), and therefore is doing no work upon the particle. In the absence of any other force, the (kinetic) energy of the particle must be conserved (so I think this is simply an application of the law of conservation of energy, yes?)

This is true only for an object rotating around one of its principle axes. In general, it is not true. The motion in general is not circular, even in the absence of external forces / torques.

But he was talking about particles only. So i think he's right; Motion of a particle when the force acting on it is continuously perpendicular to its velocity vector is circular.

 

[D H]

But he was talking about particles only. So i think he's right; Motion of a particle when the force acting on it is continuously perpendicular to its velocity vector is circular.

Two points:

1. Urmi Roy in post #95 was not talking about particles. He was talking about rotation, not revolution.
2. Classically, particles have zero angular momentum due to rotation. It is a bit silly to talk about rotational mechanics with respect to particles.

 

[sganesh88]

Two points:

1. Urmi Roy in post #95 was not talking about particles. He was talking about rotation, not revolution.
2. Classically, particles have zero angular momentum due to rotation. It is a bit silly to talk about rotational mechanics with respect to particles.

Sorry. I thought you were addressing the quoted text of Bobbybear which talked about revolution of a particle about some point; not rotation(which of course doesn't make sense as you said). And Urmi Roy is a female i guess; the tone of her messages suggests that. ;-)

 

[iitjee10]

it would be better if you start a new thread for that

 

[BobbyBear]

Heylo :)

okay with regard to post 104, thank you D H for your insights. As far as I know, the central axis is the locus of points with the smallest linear velocity (if this velocity is zero, it is termed the instantaneous axis of rotation). Thus the movement of a rigid body is the superposition of a translation given by the velocity of the points on the central axis, plus a rotation about this axis. I never thought about describing the motion as a rotation around any other axis . . . but I suppose what you say makes sense, because all movement is relative? And I'm not sure how all this ties in with the centre of mass either.. and how the equations of rotation and translation become decoupled. Anyhow, I think it's best for me to actually study the equations in order to see all this more clearly. I don't have time right now, but maybe in a few weeks . . . anyway, may I ask if you could suggest any good textbooks explaining these thigns, as I do not have any at the moment? I'll keep your posts so as to go over them again when I do get down to studying them in more depth : ) Thanks again for your insights xD


About post 106, actually I was argumenting why the energy is conserved for a particle moving in a circular trajectory, which I think was the first consideration that Urmi brought up in point No.2 of post 95 . . . and I was trying to extrapolate this reasoning to a rotating solid by argumenting that each individual particle of that solid could be viewed as a particle describing a circular translational motion, and thus the energy of each particle is conserved, and hence the energy of the solid is conserved. Correct me if I'm wrong, but it seems to me that there are only three fundamental laws of motion (at least as far as Newtonian mechanics goes), and these are the three laws that Newton put forth for particles. Solids are simply a collection of particles stuck together, and the expressions that describe/govern the dynamics of solids are merely derived from the principles put forth for particles, using the adequate maths. That is, they are not new principles, simply operative expressions that can be applied directly to finite bodies - but in the end, they simply say the same thing as applying Newton's principles to each particle of the solid and tying all these movements together would say about the movement of the solid, if you get what I mean :P

Thank you sganesh for clearing my doubt about 'torque of a solid'. But still, if you were to have a solid suspended in space, and apply only one force say at one end of the solid (not through its COM), would it rotate at all? (I'm assuming there are no physical constraints upon the solid at all - no physical pivot or anything). And about which point would it rotate? I can't seem to imagine why, if it should rotate, it should do so about any particular point.

 

[sganesh88]

Thank you sganesh for clearing my doubt about 'torque of a solid'. But still, if you were to have a solid suspended in space, and apply only one force say at one end of the solid (not through its COM), would it rotate at all? (I'm assuming there are no physical constraints upon the solid at all - no physical pivot or anything). And about which point would it rotate? I can't seem to imagine why, if it should rotate, it should do so about any particular point.

I too had that doubt a year before. I thought rotation is possible only in the presence of a pivot constraint or two opposite forces whose lines of actions are separated by a distance. But the body has to rotate if the line of action of the force doesn't pass through the COM (even if its just a single force). Why? Check whether conservation of angular momentum is conserved if the body doesn't rotate but just translates as a whole when a force acts on its tip. (Though i get a doubt. Why should angular momentum be conserved? Going to read about it now.)

Regarding the axis of rotation, draw a line from the COM to the point of application of the force; the perpendicular to this line, passing through the COM will be the axis.

 

[BobbyBear]

Regarding the axis of rotation, draw a line from the COM to the point of application of the force; the perpendicular to this line, passing through the COM will be the axis.

Hmm, if that is the axis of rotation, then there is a net torque with respect to that axis, and as you said, the 'torque' is always with respect to the COM . . .
D H said that in an inertial frame, we have: $d\boldsymbol L/dt = \boldsymbol {\tau}_{\text{ext}}$, so the solid must acquire an angular momentum with respect to an inertial frame? And also as D H said, this does not mean that it must acquire an angular velocity, as the inertia tensor is not constant in an inertial frame - though again, if the body does not rotate, then the inertia tensor would be constant, so it must rotate . . . . Hence if the body does not rotate at all, it would mean that angular momentum is conserved, no? So that can't be true if there's an external force acting on the solid with a certain eccentricity with respect to the centre of mass. Anyhow, I'm a bit dodgy with all of this, I think I need to follow some textbook that develops the general equations and explains all this with certain depth xD

 

[sganesh88]

Torque needn't always be w.r.t COM but in this case of a solid suspended in space, this is preferable.
And i know nothing about this inertia tensor thing. Yet to study about it.

 

[BobbyBear]

Torque needn't always be w.r.t COM but in this case of a solid suspended in space, this is preferable.
And i know nothing about this inertia tensor thing. Yet to study about it.

*flails* that's just the point, you can consider the torque produced by a force with respect to any point, but the motion of the solid cannot depend on which point you choose to consider to apply Newton's law of rotation . . . I think I need to study about it too *wink* if you come across any good books let me know their names xD

 

[D H]

BobbyBear:

To answer your first question regarding the decoupling of translational and rotational equations of motion, consider a system comprising a fixed number of constant mass particles. Such a system will have a constant mass and the individual components won't change (e.g., there are no chemical interactions that change the number of particles).

Denote the mass and position with respect to some inertial frame of the i^th particle as m_i and x_i. The mass, center of mass of the system, and system center of mass velocity are

$$\aligned m_{\text{tot}} &\equiv \sum_i m_i \\ m_{\text{tot}} \boldsymbol x_{\text{CM}} &\equiv \sum_i m_i \boldsymbol x_i \\ m_{\text{tot}} \boldsymbol v_{\text{CM}} &\equiv \sum_i m_i \dot{\boldsymbol x}_i \endaligned$$

The total linear momentum is the sum of the linear momentum for each particle:

$$\boldsymbol p_{\text{tot}} = \sum_i m_i \dot{\boldsymbol x}_i$$

Note that this is the total mass times the system center of mass velocity. Differentiating with respect to time,

$$\dot{\boldsymbol p}_{\text{tot}} = \sum_i m_i \ddot{\boldsymbol x}_i$$

Assume each particle obeys Newton's laws of motion:

$$m_i\ddot{\boldsymbol r}_i = \boldsymbol F_{\text{tot},i}$$

External and internal forces will act on each particle. The external forces come from outside the system; the internal forces are interactions between pairs of particles. The total force on the i^th particle is the sum of the external force on that particle plus the interactions with the other particles in the system:

$$\boldsymbol F_{\text{tot},i} = \boldsymbol F_{\text{ext},i} + \sum_{j\ne i} \boldsymbol F_{\text{int},ij}$$

Thus

$$\aligned \dot{\boldsymbol p}_{\text{tot}} &= \sum \boldsymbol F_{\text{tot},i} \\ &= \sum_i \left(\boldsymbol F_{\text{ext},i} + \sum_{j\ne i} \boldsymbol F_{\text{int},ij} \right) \\ &= \boldsymbol F_{\text{ext},\text{tot}} + \sum_{\text{pairs\,}i,j} (\boldsymbol F_{\text{int},ij} + \boldsymbol F_{\text{int},ji}$$

By the weak form of Newton's third law, interactions between particles are equal but opposite: $\boldsymbol F_{\text{int},ji} = - \boldsymbol F_{\text{int},ij}$ and thus the change in linear momentum of the system as a whole is equal to the sum of the external forces acting on elements of the system:

$$\dot{\boldsymbol p}_{\text{tot}} = \boldsymbol F_{\text{ext},\text{tot}}$$

Using the definition of the system center of mass,

$$m_{\text{tot}} \ddot{\boldsymbol x}_{\text{CM}} = \boldsymbol F_{\text{ext},\text{tot}}$$

The system as a whole obeys Newton's second law so long the particles obey the Newton's second and third laws. What about the angular momentum? The total angular momentum of the system is the sum of that of the individual particles:

$$\boldsymbol L_{\text{tot}} = \sum_i \boldsymbol L_i = \sum_i m_i \boldsymbol x_i \times \dot{\boldsymbol x}_i$$

Differentiating with respect to time,

$$\aligned \dot{\boldsymbol L}_{\text{tot}} &= \sum_i m_i \boldsymbol x_i \times \ddot{\boldsymbol x}_i \\ &= \sum_i \boldsymbol x_i \boldsymbol F_{\text{ext},\text{tot}} \\ &= \sum_i \boldsymbol x_i\times \left(\boldsymbol F_{\text{ext},i} + \sum_{j\ne i} \boldsymbol F_{\text{int},ij} \right) \\ &= \left(\sum_i \boldsymbol x_i\times \left(\boldsymbol F_{\text{ext},i}\right) + \left(\sum_{\text{pairs\,}i,j} (\boldsymbol x_i\times \boldsymbol F_{\text{int},ij} + \boldsymbol x_j\times \boldsymbol F_{\text{int},ji}\right) \\ &= \left(\sum_i \boldsymbol x_i\times \left(\boldsymbol F_{\text{ext},i}\right) + \left(\sum_{\text{pairs\,}i,j} \boldsymbol x_i \times (\boldsymbol F_{\text{int},ij} +\boldsymbol F_{\text{int},ji}) + (\boldsymbol x_j-\boldsymbol x_i)\times \boldsymbol F_{\text{int},ji}\right) \endaligned$$

The terms $\boldsymbol F_{\text{int},ij} +\boldsymbol F_{\text{int},ji}$ vanish per the weak form of Newton's third law. The final set of terms, $(\boldsymbol x_j-\boldsymbol x_i)\times \boldsymbol F_{\text{int},ji}$, vanish per the strong form of Newton's third law. Thus the total angular momentum depends only on the external forces acting on individual particles:

$$\dot{\boldsymbol L}_{\text{tot}} = \sum_i \boldsymbol x_i\times \left(\boldsymbol F_{\text{ext},i}$$


The linear and angular momenta for a rigid body can be expressed as

$$\aligned \boldsymbol p &= m \dot{\boldsymbol x}_{\text{CM}} \\ \boldsymbol L &= m \boldsymbol x_{\text{CM}}\times \dot{\boldsymbol x}_{\text{CM}} + \mathbf I \boldsymbol {\omega} \endaligned$$

When expressed in this form, the translational and rotational equations of motion decouple. The translational equations of motion take on the form

$$m \ddot{\boldsymbol x}_{\text{CM}} = \boldsymbol F_{\text{ext,tot}}$$

The rotational equations of motion were already described in post #102.

 

[BobbyBear]

THANK YOU D H,
that was very helpful xD