Simple Harmonic Motion: Simple Pendulum

[PFuser1232]

If we apply the small angle approximation so that a simple pendulum can be considered to be under going SHM, what kind of potential energy would the pendulum bob be having? If the answer is gravitational potential energy, then we have a contradiction because this would mean that the bob would rise and descend periodically in 2 dimensions but SHM according to my knowledge is a one dimensional phenomenon. If the answer is other than gravitational potential energy, we still have a contradiction because the bob can't entirely oscillate in the horizontal direction without increasing the length of the string to which the mass is attached which is complete non-sense since the string is clearly inelastic in a simple pendulum.

 

[sophiecentaur]

You are piling in on this in a rather aggressive way to poor old Physics. You should be asking where you have mis-understoof, I think, rather than claiming Physics got it wrong
The bob only has one degree of freedom - along the arc of the circle. So there is only one dimension involved. It is true, of course, that the restoring force is only proportional to the displacement for small angles and the motion is not purely sinusoidal in time

 

[olivermsun]

If we apply the small angle approximation so that a simple pendulum can be considered to be under going SHM, what kind of potential energy would the pendulum bob be having? ...

I posted a response in your previous thread. Just to reiterate: for a pendulum the "restoring force" is gravity itself, so the potential energy of the oscillator is just the gravitational potential energy mgz.

One way to formulate this problem is to pick as your single dimension θ(t), i.e., the pendulum angle as a function of time.

 

[sophiecentaur]

@MohammedRady97
If you look at what happens to a mass on a coil spring, there can also be rotation around the axis of the helix. This gives another dimension in that system too - if you want to find one.

 

[PFuser1232]

You are piling in on this in a rather aggressive way to poor old Physics. You should be asking where you have mis-understoof, I think, rather than claiming Physics got it wrong
The bob only has one degree of freedom - along the arc of the circle. So there is only one dimension involved. It is true, of course, that the restoring force is only proportional to the displacement for small angles and the motion is not purely sinusoidal in time

But if the motion of the bob along the arc is one dimensional, then circular motion is one dimensional but it's clearly two dimensional. (I'm sorry for my misunderstanding regarding the difference between a dimension and a degree of freedom if there is any, I'm still in my senior year in high school, starting university next year.)

 

[olivermsun]

"Dimension" in a physical system doesn't just have to be one the coordinates: x, y, z, t. In some systems, for example, it makes sense to use coordinates like r, radius, and angle of displacement (rotation), θ.

 

[nasu]

But if the motion of the bob along the arc is one dimensional, then circular motion is one dimensional but it's clearly two dimensional. (I'm sorry for my misunderstanding regarding the difference between a dimension and a degree of freedom if there is any, I'm still in my senior year in high school, starting university next year.)

Ok, then maybe is just a matter of a too narrow field of view, so to say.
You can have harmonic oscillations in any number of dimensions. Dimensionality is not really an issue. See for example the 3D harmonic oscillator.
The important feature of harmonic motion is the proportionality of restoring force to the displacement from equilibrium. No matter how you measure this displacement.

 

[PFuser1232]

So the displacement in this case is the angle Θ, right? But can someone help me figure out the equations for displacement, velocity and acceleration in this case? Like, i think this time it's Angular velocity, angular displacement and angular acceleration, is this true? The thing is, I'm trying to link the concept of SHM in a pendulum to other examples of SHM like a mass on a spring for instance.

 

[mal4mac]

The bob actually exists in 3 dimensions, in the Newtonian model. To simplify the problem you can (obviously) ignore the z dimension, by using (r,θ) you can ignore the r direction - like z it is a constant. That's what Sophie means by there being one degree of freedom - there is only one co-ordinate that we need to take into account (θ).

Note the dictionary definition of extension - "a measurable extent of a particular kind". So θ is as much a dimension as x or y.

 

[PFuser1232]

The bob actually exists in 3 dimensions, in the Newtonian model. To simplify the problem you can (obviously) ignore the z dimension, by using (r,θ) you can ignore the r direction - like z it is a constant. That's what Sophie means by there being one degree of freedom - there is only one co-ordinate that we need to take into account (θ).

Note the dictionary definition of extension - "a measurable extent of a particular kind". So θ is as much a dimension as x or y.

Hmmm..so, acceleration would still be a = -ω² x, and x= l theta , so a = -ω² l theta?

 

[PFuser1232]

And the potential energy would be mgx? Shouldn't x be vertical then? I'm confused.

 

[xAxis]

There is no vertical. Everything happens along the (small) arc of the circle. Only angle is needed to determine position, so everything is a function of that angle.

 

[PFuser1232]

There is no vertical. Everything happens along the (small) arc of the circle. Only angle is needed to determine position, so everything is a function of that angle.

How would we have gravitational potential energy if there were no vertical?

 

[sophiecentaur]

How would we have gravitational potential energy if there were no vertical?

Have you been reading what's been written about the fact that the co ordinate system you use, is arbitrary? Of course the PE changes with angle but angle is the best variable to analyse the motion with. You need to learn to be flexible about these things and not try to prove to yourself that the textbooks have been getting it wrong. They haven't; you are just not looking at things in the best way to get a (valid) answer.