What is Radius of gyration: Definition and 29 Discussions
Radius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated there.
Mathematically the radius of gyration is the root mean square distance of the object's parts from either its center of mass or a given axis, depending on the relevant application. It is actually the perpendicular distance from point mass to the axis of rotation. One can represent a trajectory of a moving point as a body. Then radius of gyration can be used to characterize the typical distance travelled by this point.
Suppose a body consists of
n
{\displaystyle n}
particles each of mass
m
{\displaystyle m}
. Let
r
1
,
r
2
,
r
3
,
…
,
r
n
{\displaystyle r_{1},r_{2},r_{3},\dots ,r_{n}}
be their perpendicular distances from the axis of rotation. Then, the moment of inertia
I
{\displaystyle I}
of the body about the axis of rotation is
I
=
m
1
r
1
2
+
m
2
r
2
2
+
⋯
+
m
n
r
n
2
{\displaystyle I=m_{1}r_{1}^{2}+m_{2}r_{2}^{2}+\cdots +m_{n}r_{n}^{2}}
If all the masses are the same (
m
{\displaystyle m}
), then the moment of inertia is
I
=
m
(
r
1
2
+
r
2
2
+
⋯
+
r
n
2
)
{\displaystyle I=m(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})}
.
Since
m
=
M
/
n
{\displaystyle m=M/n}
(
M
{\displaystyle M}
being the total mass of the body),
I
=
M
(
r
1
2
+
r
2
2
+
⋯
+
r
n
2
)
/
n
{\displaystyle I=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
From the above equations, we have
M
R
g
2
=
M
(
r
1
2
+
r
2
2
+
⋯
+
r
n
2
)
/
n
{\displaystyle MR_{g}^{2}=M(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
Radius of gyration is the root mean square distance of particles from axis formula
R
g
2
=
(
r
1
2
+
r
2
2
+
⋯
+
r
n
2
)
/
n
{\displaystyle R_{g}^{2}=(r_{1}^{2}+r_{2}^{2}+\cdots +r_{n}^{2})/n}
Therefore, the radius of gyration of a body about a given axis may also be defined as the root mean square distance of the various particles of the body from the axis of rotation. It is also known as a measure of the way in which the mass of a rotating rigid body is distributed about its axis of rotation.
Hello,
I'm given a problem of a mass rolling down an incline with mass 'm', radius 'r', and radius of gyration Rg, and I need to write the Lagrangian for the motion. I'm confused on why both r and Rg are given. Don't I just need one of the two for the moment of inertia?
Thanks!
Homework Statement
Homework Equations
Centre of gravity: X=m1x1-m2x2/m1-m2
MOI rectangle: 1/3ml^2
MOI triangle: 1/18md^2
Radius of gyration: Ixx=mk^2
The Attempt at a Solution
Mass of body 1: b*l*p = 0.8*1*10=8kg
Mass of body 2: 1/2b*h*p = 1/2(0.4)*0.6*10=1.2kg1.1
X=m1x1-m2x2/m1-m2...
Find the moments of inertia about the x-axis, y-axis and the origin. Also, find the radius of gyration about the x-axis and y-axis.
y = 0, y = b, x = 0, x = a
Rho = ky
1. Is ky the density function?
2. Do I integrate over dxdy or dydx?
3. Are the limits of integration y = 0, y = b, x = 0...
I was not sure if this was the best place for this, it could fit here, in the Chemisty section or the Programming section. So feel free to move if needed.
Essentially I have been modeling polymers in python and using a Monte Carlo, Metropolis type algorithm, to minimise its energy into...
Homework Statement
A 30 kg wheel has a center of mass 0.1 m left from the center of the wheel and radius of gyration KG = 0.15 m. Find the angular acceleration if the wheel is originally at rest. The radius of the wheel is 0.25m.
Homework Equations
I=mk^2
T=f*d
M=I*a
Fn acting bottom in Y...
Hi There
I am doing a little test program for some tire testing and just need to make sure I am doing something right, I am calculating the radius of gyration of my tire and wheel separately using the formulas from Dunlop...
Homework Statement
the theory behind Parallel axis theorem
Homework Equations
parallel axis theorem:Io=Ig+md²
Radius of Gyration=Ig=m*K²
The Attempt at a Solution
ok folks If I understand this theory correctly the radius of gyration is the radius or distance from the axis of rotation to the...
Homework Statement
Stacy is throwing a discus. During the throw, she applies an average torque of 90Nm to the discus for 0.3 seconds. The discus has a mass of 1.0kg, and has a radius of gyration of 0.1m about its spin axis. If the initial angular velocity of the discus was zero, what was the...
Homework Statement
The Attempt at a Solution
Alright, so the solution manual shows that the intertia of the disk is:
I = m * k2, where k is the radius of gyration
However why can't the inertia of the disk be:
I = 0.5 * m * r2, where r is the radius of the disk
This is the formula for the...
Homework Statement
Homework Equations
radius of gyration:
r = root (I/m)
I = moment of inertia
m = mass
parallel axis theorem given above
The Attempt at a Solution
Okay, so I think the moment about CM is just m*0.24^2, but after that, I'm less sure.
Is the moment about the hip just...
Hey,
If initially I have some solid sphere spinning at some initial angular velocity and in its final state I have the same solid sphere spinning at a different angular velocity except some of its mass has moved to a ring 45 degrees in latitude from centre , such that this ring of mass is...
Out of many properties polymer scientists are interested to calculate one of the most common is "Rg" i.e. Radius of Gyration. Can anyone put more light on the physical significance of this value?
Can Rg value of two polymers be compared? If yes what conclusion can be drawn from such comparison?
Hey Guys,
I having trouble with understanding radius of gyration, could someone please explain what it is? I have just never understood it's full meaning. So for example, the radius of gyration of a spinning wheel of a car is ...some value... What does that mean?
Thanks
I need to calculate the radius of gyration for a generic, convex polygon, where the density is constant, the axis of rotation is the centroid (which is known), and the positions of the vertices are known. Does such an equation exist?
Homework Statement
What is the radius of gyration (in meters) for the steel flywheel shown? The width of its rim, L, is as given below. The density of steel is 7500 kg/m3. The outside diameter (OD)for the wheel is 2000 mm, and the inside diameter (ID) is 1840 mm as shown in the figure. The...
Homework Statement
Determine the polar moment of inertia and the polar radius of gyration of the shaded area shown with respect to point P.
http://imgur.com/8Kc1S
Homework Equations
Jp = Ix + Iy
Ix = &int y^2dA
Iy = &int X^2dA
The Attempt at a Solution
A = 2(a/2)(a) +...
Homework Statement
Homework Equations
i guess its r=sqrt(I/A)
where A is the area of the circle thing and I is the moment of intertia.
The Attempt at a Solution
I guess I'm just having trouble getting I. A is 33pi
Homework Statement
By using spherical coordinates, find the radius of inertia (Is this the same as the radius of gyration?) about the z-axis of the constant density solid which lies above the upper half of the cone x2 + y2 = 3z2 and below the sphere x2 + y2 + (z-2)2 = 4. For a constant...
I'm trying to figure out the radius of gyration of the frame about O.
Homework Statement
A rectangular frame is put together with massless rods having identical 1.8 lb weights placed at the corners as shown in the figure. The frame is pivoted about an axis passing through O, the center of...
1. A solid disc of radius r, is rolling down a variable incline (a ramp). Show that the acceleration of the centre of mass, C is given by:
a=g[ sinθ – F/Ncosθ ].
I have done this as shown below.
N is the normal reaction and F is friction.
N = mgcosθ
F = µN = µmgcosθ
mgsinθ – F = ma...
Hi. I'm having trouble knowing where to start on a problem. Basically, I have to find the radius of gyration of this diamond shape, about an axis through its centre and perpendicular to the plane. All sides are 0.5m in length and from tip to tip horizontally it measures 0.8m. I'm not quite sure...
Radius of Gyration? REALLY need help ASAP!
Hey -
If any of you can help me with the following problem asap that would be awesome!
A racquet consists of uniform lamina that occupies the region inside the right-hand loop of r^2 = cos 2theta on the end of a handle (assumed to be of...
I'm learning how to find these things in school but I have no idea what they are. Moment of inertia's units are distance^4 such as in^4 or mm^4. Mass moment of inertia has units of mass*distance^2 such as kgmm^2. Radius of gyration is a distance such as mm or in.
So what are they exactly and...