Radius of Gyration

[97BlackBetty]

I am working on physics and I just simply do not understand a question.

The question is: Why is the radius of gyration of any wheel always smaller than the wheel's actual radius?

Gyration is defined as:
1. To revolve around a fixed point or axis.
2. To move in spiral or spirallike course.
3. To oscillate or vary, especially in a repetitious pattern.

And as a verb:
1. To turn round and round in a circle.

 

[MAKG]

The "radius of gyration" is the RMS distance of every part of the object from its axis. It's closely related to the moment of inertia.

I'd suggest putting the dictionary down. Colloquial meanings will only confuse the issue.

Translate the first sentence to integral form as a first step.

 

[AngryLepercon]

My guess would be less movement of the wheel! kinda like a sprocket on a dirtbike! Example if the sprocket is smaller then the tire or wheel then the sprocket will turn at triple the rate of the wheel. If i was the same size then the sprocket and the wheel would be turning at the same time at the same speed! My guess it would be less movement of important parts! I'm prob wrong but i was bored and tired lol.

 

[97BlackBetty]

Alright so as I understand it, "radious of gyration" is the greatest radial point in which a given force applied to an object will not overcome the moment of inertia?

 

[MAKG]

I'm afraid that doesn't make much sense.

In a frictionless system, any force applied any distance at all (even infinitesimally) off axis will cause the object to accelerate. This is Newton's 2nd Law. "Overcoming the moment of inertia" doesn't parse.

I'll suggest that this forum is not a good place for this sort of question -- most of what you get told will be wildly wrong, as most folks here have no training at all in rigid body physics, and don't even understand what the terms mean (most folks will confuse angular velocity with angular momentum, for instance, and even worse, power with torque). Perhaps you can contact your TA or instructor? Diagrams and hand drawings can help substantially here. And these folks (unlike the board members) will have some idea about what has been introduced in the course....

 

[Chris.S.]

My cat's breath smells like cat food.

 

[Sunk]

My guess would be less movement of the wheel! kinda like a sprocket on a dirtbike! Example if the sprocket is smaller then the tire or wheel then the sprocket will turn at triple the rate of the wheel. If i was the same size then the sprocket and the wheel would be turning at the same time at the same speed! My guess it would be less movement of important parts! I'm prob wrong but i was bored and tired lol.

That's actually incorrect. A single point on the sprocket will revolve around the axle at a slower rate than a fixed point on the tire. The distance from the axle to the point on the tire is greater than the distance to point on the sprocket, therefore the distance it must travel is greater, and it's impossible for them to move at different speeds.

 

[97BlackBetty]

I'm afraid that doesn't make much sense.

In a frictionless system, any force applied any distance at all (even infinitesimally) off axis will cause the object to accelerate. This is Newton's 2nd Law. "Overcoming the moment of inertia" doesn't parse.

I'll suggest that this forum is not a good place for this sort of question -- most of what you get told will be wildly wrong, as most folks here have no training at all in rigid body physics, and don't even understand what the terms mean (most folks will confuse angular velocity with angular momentum, for instance, and even worse, power with torque). Perhaps you can contact your TA or instructor? Diagrams and hand drawings can help substantially here. And these folks (unlike the board members) will have some idea about what has been introduced in the course....

Yea, I tried learning it from the teacher several times and I keep failing to learn it. I was honestly hoping that it was just something stupid that I was not putting two and two together.

 

[Mark_88]

I thought this was another boobie in motion thread...rats!

 

[AngryLepercon]

That's actually incorrect. A single point on the sprocket will revolve around the axle at a slower rate than a fixed point on the tire. The distance from the axle to the point on the tire is greater than the distance to point on the sprocket, therefore the distance it must travel is greater, and it's impossible for them to move at different speeds.

I always thought that the sprocket will spin say 3 times before the tire makes one rotation! smaller the sprocket the better acceleration. The larger the sprocket the more top end. O well i'm stupid lol

 

[Sunk]

Well you are right about the smaller sprocket giving better acceleration, but your thinking was a little backwards I guess. It goes back to what I said before about the larger radius at specific RPM equaling higher speed. The chain goes around the outside of the out sprocket much as the ground contacts the outside of the tire. When you increase the radius of the sprocket, it's like putting bigger tires on it.

It's like deeper gears in a car. With 5.13s you'll have awesome acceleration, but will spinning high RPM's once you're at speed. With 3.55 gears you'll have more sluggish acceleration, but will able to go faster before reaching high engine RPMs..

If that makes sense.... I'm having trouble figuring out the words to explain it.

 

[dennis461]

I'll give this a try.
The radius of gyration R is an imaginary point on an object.
A slender rod length L and mass M has R=.289L
=================ROD================
._____.289L _________C___________________L
If you melted the rod and made two small metal balls (Engineering takes balls)
Connected the two balls with two weightless rods of length .289L,
The mass would have the same radius of gyration, rotational inertia would match the rod length L
O--------------------O
._____.289LC
.
Now consider a clock pendulum.
Most of these have a large mass at the end, a lightweight rod.
The center of gravity of the pendulum is near the center of gravity of the weight on the end.
The radius of gyration would be the distance from the fulcrum to the center of gravity, thus ALWAYS shorter than the entire pendulum.

 

[Bray D]

Wow, how did we spawn discussion of chain drive systems from the original post? lol.

Anyways, this was my understanding of the radius of gyration. The radius of gyration is the distance from the physical center of an object (wheel) and its actual center of mass. This is why we have our tires/wheels balanced, to bring the center of mass back to the center of the wheel.

For any perfectly round object (assume uniform thickness), the cause of the center of mass to be different from the center of the circle, is a non uniform density. At a distance further from the center than the radius of the circle itself, there is no mass. Therefore the center of mass cannot be located there.

 

[Joe400]

I learned about radius of gyration a few semesters back and if i recall its--- the radius of the object (the distance from the center point) where if all the mass of the object could be located at that point, and rotating(spinning)the inertia would be the same as that of the original object(with its uniformly distributed mass) if it were spinning.

I found a picture

I had a hard time with that unit. good luck

 

[low_five]

mass towards the center of the axis will be 'easier' to spin.
mass further from the axis will be 'harder'

if you take a disk of metal and melt the disk down and create a ring with the same radius as the disk you will have moved 'easy to move' material to the outside of the disk. the material will have been moved from 'easy to move' areas to 'harder to move' areas. if they have same radius, the ring shaped object will be more difficult to spin every time.

therefore it must be smaller if it is to be equally as easy to spin as the disk.