To find the number of degrees of freedom: (i)a rigid body fixed at a point (ii) a rigid body rotating about a fixed axis (ii) two particles connected by a massless spring Attempts: (i) Suppose,we are in a fixed co-ordinate system O'.The rigid body is fixed at a point O within it. So, we can think of the body as-"hanging in the air from O". To locate any point P in the body,O'P=O'O+OP. Now O'O is known;P has three co-ordinates relative to O,but with constraint OP distance fixed.Hence, degrees of freedom=2. However, this is not the last word. Since, we are talking about a rigid body,we must also consider the space orientation.So, there will be three more space orientation co-ordinates for the body. So, number of degrees of freedom=2+3=5 (ii)Here also, it looks that CM has two independent co-ordinates;and there are additional three space orientation co-ordinates.So, Degrees of Freedom=2+3=5 (iii) since the spring is massless, it cannot bend.So, only motion possible is along a linear direction.This is a system of two particles and we may think of CM motion.Clearly, there is only one degree of freedom. Please check if the above analysis is correct and suggest possible improvements.
Hmm. It's a bit of a tricky question, but I came up with a different answer. Generally I think of degrees of freedom as things that are allowed to change, or equivalently, the coordinates you need to describe something. So, for the first question, the length of the rod is fixed. But the polar and azimuthal angles are allowed to vary. So I think the answer to the first one should be 2. (Again, please correct me if my definition is different from yours.) For (2), I would use the same logic. You always know where both of the centers of mass of the rigid bodies are whenever you know the information about the center of mass. So I would say that there are two degrees of freedom here, too. For (3), you need three degrees of freedom, because the center of mass is allowed to move. You'll need the location of the center of mass (3). Now, given the location of the center of mass, you should be able to find the distances of each of the particles. I think you'll also need a vector which tells you where each particle is in relation to the other, which is three more deg. of freedom. So for this one I think you need 6. If I screwed anything up, which is probable, let me know. In particular, I have never heard this before. If this is true, then you only need the position of the center of mass, which can be had with three coordinates.
From what I understand, unrestricted rigid bodys have 6 degrees of freedom (3 translational plus 3 rotational, one for each spatial axis) While particles have 3 translational. i) I interpreted it's location as being fixed, and thus not having the 3 translational DOF, leaving 3 rotational degrees of freedom preasant. ii) This one is confusing: what does a 'fixed-axis' mean exactly (if an axis is fixed I would think that the body could NOT rotate about it...) Perhaps: It can only rotate about the 1 axis, (1 DOF) + if it can translate, add another 3 = 4 DOF. iii) Since this is a linear 2 particle system, we can rotate it allong 2 axis, and translate it 3, giving 5 DOF, plus 1 for the distance between the particles = 6 DOF. -Andrew
I agree: 3 DOF. A fixed point removes 3 DOF (question 1). A fixed axis removes two more, so only one DOF here. Think of a globe of the Earth with its base fixed to a table. All the globe can do is spin about one axis. One parameter describes the state of the globe, so one DOF. Bingo.
Why three for the first one? I would expect that it be two degrees of freedom---set the origin at the fulcrum, and you only need theta and phi (polar and azimuthal angles) to specify the orientation.
Think about it this way: Those two angles define an axis. The body is still free to rotate around that axis.
Thank you everyone...I was not accessing this site for couple of days...so, I could not join the discussion... I want to modify my answers... (i)As the rigid body is fixed at a point,we have automatically 3 degrees of freedom reduced.Normally a rigid body has 6 DoF.So,we get 6-3=3 DoF.No additional constraints are imposed.So,DoF=3 (ii)The answer is one.A rigid body rotating about an axis is not only fixed at a point,also its orientation is fixed (since it is rotating about only one axis). Among three Eulerian angles, we have 2 angles=constant by this 2nd constraint.So,only \(\phi\) motion is possible. (iii)If you think in one dimension this would involve two degrees of freedom. The spring cannot introduce any geomrtical constraints.
(i) Good. (ii) The question is not particularly well-phrased. Both a bead on a wire (two DOF) and a globe on a desk (one DOF) can be viewed as constituting a rigid body rotating about a fixed axis. Which answer is "right" depends on the instructor (or on his answer book). (iii) The spring most certainly adds a degree of freedom. Andrew nailed the answer: Three DOF for translation (the spring connects the two particles; they are free to move about as a unit), plus two DOF for rotation (rotation about the axis defined by the line between the particles doesn't count), plus one DOF for stretching and compressing the spring. Think of a diatomic gas such as hydrogen. Below around 100K, hydrogen has a specific heat ratio of about 5/3, the same as for a monoatomic gas. The bond is so tight that the molecule acts like a point mass at low temperatures. It essentially has three degrees of freedom. An ideal gas with three degrees of freedom has a specific heat ratio of exactly (3+2)/3 = 5/3. From 100K to 250K, hydrogen's specific heat ratio rises to about 7/5. The bond has loosened up enough that energy can be partitioned into rotation as well as translation: Five degrees of freedom. An ideal gas with five degrees of freedom has a specific heat ratio of exactly 7/5. Above 750K hydrogen's specific heat ratio begins to rise again. At these temperatures the bond has loosened significantly so the rigid rod model used for the moderate temperature range is no longer valid. Energy can be partitioned into stretching and compressing the bond as well as into translation and rotation. An ideal gas with six degrees of freedom has a specific heat ratio of exactly 8/6 (or 4/3). However, molecular hydrogen dissociates before its specific heat ratio reaches 4/3. It comes dang close, however.
(ii) Agree! But,in my opinion,I will call that bead as a smaller globe.Once you call it bead, you are indicating that it is a point mass and a point mass cannot spin. (iii) I do not think spring,in the given language of the problem, will contribute anything to DoF of the system.Note that I put down the spring is massless.
(ii) Call it a big fat globe on a wire, then. The axis is still fixed. The issue is whether there exist some freedom to translate along the fixed axis or just rotate about it. (iii) That is the typical idealization of a spring (massless, that is). Such "massless" springs still have a spring constant: They can stretch and compress. If the spring was to play no part the question should have read as a massless rod rather than a spring.
(ii)But as the question says the rigidbody is rotating around an axis,I think we may safely disregard the possiblity of translation along the direction of axis. If it translates,I have no problem to add more DoF to the aanswer. (iii) DoF=3N-m where N is the no of particlesand m is the number of constraints. If you had a rigid rod in place of a spring, you would get fixed separation. That is L=constant.This matters,as the two particles cannot at all move independent of each other---thus reduces the possiblity of another DoF. The spring can stretch or compress.But that does not give you a geometrical constraint---you do not even have in hand the functional dependence of the separation.So, even one particle goes to the right,the other can move,at least to some extent,to the left.This gives you an additional DoF.
Read this article: http://en.wikipedia.org/wiki/Degree...mistry)#Example:_classical_ideal_diatomic_gas It makes the exact same argument as did Andrew: 3 DOF translation, 2 DOF rotation, 1 DOF vibration, total of 6.
As far as my knowledge (graduate mechanics course) goes,DoF has nothing to do with momentum---as indicated in Wikipedia. DoF of a particle (point) without any constraint is three and for a system of N particles is 3N-m where m is the number of constraints. DoF of a rigid body without any constraint can be obtained from this formula=6.Here,3 DoF corresponds to the motion of CM of the body and 3 other DOF corresponds to the orientation of the body in space specified by Eulerian angles. Personally,I feel wikipedia or hyperphysics---they are good to some extent, but I do not take them as final word.
Even then: if the motion is considered along X axis, we have one DoF.No rotation and one vibration.Total=2
The problem simply states that the particles are connected by a spring. What are the constraints on motion? What in the problem statement makes you think motion is constrained to one axis or that the particles cannot rotate (as a group)?
If you want to orient the co-ordinate axes such that the motion is in 3D, you will have three translational degrees of freedom for each of the particles.That is simply a matter of test.I made my axes such that the spring lies in the X direction.Then,in my system, each of the particle has one DoF.Total is two.This automatically considers the vibratory motion. If I orient the axes such that the motion is in 3D,I would have 3 DoF for each of the particles.Total six. Rotation would involve torque/twist across the massless spring giving rise to infinite angular acceleration.That is forbidden.After all, there is no sense of rotation of the point masses attached to the spring. Practically, there is NO constraint in this problem.So, DoF=2X3-0=6 if the motion be visualized in 3D.
