# Is simple harmonic motion also a pure translatory motion?

• B
• tbn032
Hi,I think that in a straight line in which there is no rotation of the object, the object would be considered as a pure translational motion.

#### tbn032

Is simple harmonic motion also a pure translatory motion?"A rigid body moves in pure translation if each particle of the body undergoes the same displacement as every other particle in any given time interval" [Halliday and Resnick, Physics].If not,then how does shm deviate from this definition>

tbn032 said:
Is simple harmonic motion also a pure translatory motion?"A rigid body moves in pure translation if each particle of the body undergoes the same displacement as every other particle in any given time interval" [Halliday and Resnick, Physics].If not,then how does shm deviate from this definition>
Consider a pendulum bob (on a cycloidal shaped track if one is concerned about linearity). It obeys simple harmonic motion. But is its motion pure translation?

vanhees71 and bob012345
One way to answer @jbriggs444 excellent question is to draw the pendulum bob and track the paths of two different points on the bob.

I think the pendulum on cycloidal shape track would not be pure translational as different point will have different velocity at same instance.but if about the harmonic motion in a straight line(e.g box attached to a spring on a horizontal surface)will it be pure translational?

bob012345 and jbriggs444
tbn032 said:
I think the pendulum on cycloidal shape track would not be pure translational as different point will have different velocity at same instance.but if about the harmonic motion in a straight line(e.g box attached to a spring on a horizontal surface)will it be pure translational?
Great question! What do you think?

Also, welcome to PF!

I think it will be a pure translational motion since velocity of each particle is in same direction and has same magnitude.

bob012345
tbn032 said:
I think it will be a pure translational motion since velocity of each particle is in same direction and has same magnitude.
Ok, but relate this back to the original definition you posted.

tbn032 said:
I think it will be a pure translational motion since velocity of each particle is in same direction and has same magnitude.
Doesn't the bob rotate at all? (In practice or even in theory)

sophiecentaur said:
Doesn't the bob rotate at all? (In practice or even in theory)
If you chase back, I think you'll find that he's switched scenarios to a non-rotating block sliding on a surface due to a spring.

jbriggs444 said:
If you chase back, I think you'll find that he's switched scenarios to a non-rotating block sliding on a surface due to a spring.
I was wondering how that could be achieved, though. I now see he seguéed from one to the other. Basically, a mass on a spring is 'perfect' SHM as long as the spring is long enough. I think a hair spring on a wristwatch is one of the best (rotational) SHMs around.

jbriggs444
bob012345 said:
Ok, but relate this back to the original definition you posted.
According to the original definition I posted, the object is in pure translational motion if each particle of the body undergoes the same displacement as every other particle in any given time interval. If an object is in SHM in straight line(box attached to a spring in horizontal) the overall displacement of the object is zero but between any time period, each particle of the object show the same displacement(whether it is zero or between -x and x).(x is the displacement of the object with respect to the initial position of the object where the object connected to the spring is at rest)

bob012345
Hence I think the object is in pure translational motion.

bob012345
tbn032 said:
Hence I think the object is in pure translational motion.
That's if you define the displacement as translational. But the displacement can be rotational or a combination of the two as long as the restoring force/torque is still proportional to displacement (by an appropriate definition).
I must say, I haven't understood why the idea of translational seems to be so important to you. Could it be a search for 'classification'?

Systems like pendulae are just not examples of pure SHM but there are examples of SHM, involving rotation (hairspring in a wristwatch escapement) . In systems involving a combination of translational and rotational motion then the 'displacement' quantity may be more complicated.
Maybe a better (more universal) definition of SHM would be that the variation of Potential Energy (or KE) should be sinusoidal with time. Would that allow you to 'classify' thing in a satisfactory way?

bob012345
sophiecentaur said:
That's if you define the displacement as translational. But the displacement can be rotational or a combination of the two as long as the restoring force/torque is still proportional to displacement (by an appropriate definition).
I was not talking about displacement in SHM in general, I was talking about whether an SHM in a straight line in which there is no rotation of the object can also be considered as pure transitional motion.(for e.g a object which is connected to a spring kept in the horizontal surface which is displaced parallel to the surface)

tbn032 said:
I was not talking about displacement in SHM in general, I was talking about whether an SHM in a straight line in which there is no rotation of the object can also be considered as pure transitional motion.
Glib answer: you asked if an object you defined to be in pure translational motion is in pure translational motion. What do you think?

More detailed answer: nothing is perfectly rigid. If the pendulum bob is a long thin piece of elastic with its length perpendicular to the motion then the ends of it will whip back and forth as it moves. If we replace the elastic with something slightly stiffer the whipping will reduce. If we replace that with something even stiffer the whipping will reduce further. But the whipping effect will always be present (in tiny amounts) in any bob that is not infinitely rigid or infinitely small - which means, in any real material.

So the best answer is: you should not care about this question. The question you should care about is that, given that there must be some flex in any real system, is the energy involved in that flex negligible? If you are trying to analyse a spherical steel bob weighing a kilogram or two on a spring such that the period is a second or two, then flex in the bob is utterly irrelevant and you can idealise it as pure translation. If you've got a bob made of rubber with a spring constant similar to your actual spring, you need to worry about that.

bob012345 and sophiecentaur
Ibix said:
Glib answer: you asked if an object you defined to be in pure translational motion is in pure translational motion. What do you think?
I was concerned about the fact that since the object is in SHM,it will not be considered a pure translational motion.so far I have concluded that object being in SHM in a straight line in which there is no rotation of the object can also be considered as pure transitional motion.is my conclusion correct?

bob012345
tbn032 said:
is my conclusion correct?
It depends how precise you need to be. In reality you are wrong, because the bob will flex by a tiny amount. But you would need fairly high quality lab gear to detect the flex if the bob is metal and the motion isn't too violent, so except under fairly extreme circumstances you would be totally fine approximating the motion as purely translational.

I'm sorry to be awkward, but this is a fundamental problem with an approach to physics that tries to fix exact classifications of particular behaviours of reality. The answer is always either "you defined the model that way so obviously yes" or "reality is more complex than that so maybe, sometimes, and it depends how closely you look".

In an idealised model where there is absolutely no flex in your bob and you've declared the motion to be purely translational then it is purely translational as a matter of tautology, whether it's harmonic or not. Reality is not an idealised model, though, and no motion under acceleration is purely translational. The question is, is a model where the simple harmonic motion is purely translational close enough to whatever reality you are studying to be useful?

bob012345
tbn032 said:
I was not talking about displacement in SHM in general, I was talking about whether an SHM in a straight line in which there is no rotation of the object can also be considered as pure transitional motion.(for e.g a object which is connected to a spring kept in the horizontal surface which is displaced parallel to the surface)
Isn't the definition a part of that question? How else could the motion be described under those conditions? (Unless you leave the ideal model and talk about a squidgy object.)

bob012345
tbn032 said:
I was concerned about the fact that since the object is in SHM,it will not be considered a pure translational motion.so far I have concluded that object being in SHM in a straight line in which there is no rotation of the object can also be considered as pure transitional motion.is my conclusion correct?
Yes.

tbn032 said:
I was concerned about the fact that since the object is in SHM,it will not be considered a pure translational motion.so far I have concluded that object being in SHM in a straight line in which there is no rotation of the object can also be considered as pure transitional motion.is my conclusion correct?
I still have not grasped what you are getting at, apart from an attempt at classification.

If the object on the spring is attached at its CM then there is no cause of rotation. Once you introduce anything else into the model then you will have a multiple oscillating system - a compound pendulum with more than just one mode - additional time variations.

sophiecentaur said:
I still have not grasped what you are getting at, apart from an attempt at classification.

If the object on the spring is attached at its CM then there is no cause of rotation. Once you introduce anything else into the model then you will have a multiple oscillating system - a compound pendulum with more than just one mode - additional time variations.
I believe what @tbn032 was getting at in the OP was if SHM back and forth along a line was still considered pure translation according to the definition in Halliday and Resnick. The answer reached was yes.

bob012345 said:
I believe what @tbn032 was getting at in the OP was if SHM back and forth along a line was still considered pure translation according to the definition in Halliday and Resnick. The answer reached was yes.
In addition, if one hangs a long, rigid, horizontal bob beneath two parallen pendulums then even though the oscillation is not in a straight line, it is pure "translation" because there is no "rotation".