Centre of Mass & SHM: Help Understanding & Graphing

[nic0la]

Hi,
I have a hard time trying to understand the Centre of Mass as well as the graphing for SHM.
Please help. Thanks heaps.

 

[sunilkamadolli]

center of mass is the point where all the mass of the object is concentrated. This can help make calculations easier. You can even use Newton's lawz on systems of particles...center of mass of objects...i don't know wut SHM iz...wut does it stand for?

Hope that helpz
- Mr Kamadolli

 

[outandbeyond2004]

Well, sunny, not exactly correct. The center of mass is the point at which any force can move the object without rotating it. If you apply a force to the object at another point, and the object is free to move any way, it will rotate some.

Graphing for the Simple Harmonic Motion? Please rephrase the question.

 

[arildno]

outy, that is incorrect.
As long as the direction of the force is parallell to the vector connecting C.M. and the point at which the force acts, the object will not rotate.

The position to Center of Mass is found by the making a weighted average
of the particle positions, over all particles the object consists of.

The weights are the particles' own masses.

 

[arildno]

nic0la, with graphing of SHM, do you mean to find the curves in the phase plane?

 

[jdavel]

nic0la,

As to your question about the center of mass:

sunilkamadolli's answer: dead wrong

outandbeyond2004's: close

arildno's anwer: right.

An intuitive way to think about CM is to imagine balancing an object on one finger.

Start with a flat object, say a thin uniform disk lying flat, your finger will have to be at the center; that's the CM. Same for a yardstick. With complicated shapes (say a map of the US) it's harder to guess where your finger would have to be. But the physics will work out, and there's some point where it will balance; that's the CM. By the way there's a city somewhere in Kansas (I think) that claims to be at the center of mass of the US.

With solid objects, it's a little trickier, and this is where outandbeyond2004 and arildno's answers disagreed. The CM of mass is usually somewhere inside the object, so you can't really touch it. Now the condition for balancing is that the CM be directly above (or below, in some unusual cases) the point where your finger holds the object.

As to graphing SHM. In physics (at least classical physics) a graph of the motion of an object usually means a graph of its location as a function of time, that is x vs t. Simple harmonic motion is a special case of a more general kind of motion where an object moves back and forth in some regular, periodic way. That is, the object oscillates. So the graph of x vs t (t along the horizontal direction) is a zigzag line going back and forth across the t axis. In SHM that zigzag line has the shape of a sine curve.

Two interesting facts about SHM:

1) The graph of velocity vs time and acceleration vs time are also sinusoidal. This isn't true with all oscillatory motion, and it's what makes SHM special. and relatively easy to do calculations with.

2) The oscillations of lots of things in nature (the swinging weight on a pendulum, objects on springs, a cork going up and down on waves in the ocean, even atoms in crystals, etc.) are pretty close to SHM.

 

[arildno]

Just a minor comment here:
You cannot balance a doughnut by putting your finger on C.M, because in this special case, the C.M. does not lie in the doughnut at all, it is at the empty centre!
(You are certainly able to touch C.M., though).
You must align the doughnut along the vertical, and then use jdavel's method by touching the doughnut at a point either directly above or below C.M.

 

[sunilkamadolli]

ok...sorry...i didnt quite get that...
as arildno says...the center of mass does not have to lie on the object...i should have said it is the location where that point "acts" like all the object's mass is concentrated...and all the external forces are applied...i think
note - for a uniform symetrical solid body the center of the mass would be itz geometrical midpoint.
when considering a solid object whose atoms are stationary and there are a lot of atoms...we can use calculus to find center of mass...

 

[krab]

The center of mass is the point at which any force can move the object without rotating it.

arildno and jdavel, read out..'s definition again. (Clue; note the word "any".) It is not only correct, but the most concise one I've seen. I like it.