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What is the Moment of inertia of a cone rotating around its axis of rotational symmetry?
Wiki User
∙ 14y ago
The moment of inertia of a cone about its central axis, start with the standard Intertia equation I = integral r^2 dm dm = rho dV (rho is density) (dV is basically volume) dV = r dr dtheta dx not going to prove that here but you will see in the integral that this does indeed form the volume. integral will be refered to as int from here on. This now forms the triple integral I = rho int(0 to H) int(0 to 2pi) int(0 to r) r^3 dr dtheta dx solving the integral leaves I = rho int(0 to H) int(0-2pi) 1/4 r^4 dtheta dx solving the second integral leaves I = rho int(0 to H) 1/2 pi r^4 dx ok so now you have to sub in the equation for r (the radius) of the cone r = (R/H)x this is the radius at the base divided by the height of the cone multiplied by the distance along the x axis. this equation gives you r at any point this gives you I = rho int(0 to H) 1/2 pi [(R/H)x]^4 dx time to do some housekeeping and take all the constants outside the integral I = (rho pi R^4) / (2 H^4) int(0 to H) x^4 dx this can now be solved and simplified to I = (rho pi R^4 H) / 10 At this stage your solution is complete, however you can tidy up the equation by taking out the mass term. m = (rho pi H R^2) / 3 split the Inertia term up to serperate out the mass term I = [(rho pi H R^2) / 3]*[ (3R^2)/10 ] this is now the complete solution in terms of mass I = (3mR^2)/10 I hope this manages to help some poor unfortunate student who gets set this question.
Wiki User
∙ 16y ago
Wiki User
∙ 15y agoAssuming the axis coordinate system is such that z describes the vertical axis, y is the horizontal axis and x being the axis coming out of you computer screen then the moments of intertia are as follows (r=radius, h=height, m=mass): Ixx=Iyy=(3/80)m(4r2+h2) Izz=(3/10)mr2
Wiki User
∙ 12y agoDepends on the axis considered.
For the axis through the center perpendicular to the plane of the disk:
I = m*r^4 /2, where r is the radius of the disk.
For an axis through the center on the plane of the disk:
I = m*r^4 /4
Wiki User
∙ 14y agoThe derivation for this is somewhat long, so I will not run through it, but the answer is 3/10 MR2.
Wiki User
∙ 12y agoMoment of inertia of hollowsphere = 2/5 M(R^5-r^5)/(R^3-r^3)
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What physical characteristic does the moment of inertia of a rotating object most directly and accurately measure?
This is rotational inertia. When inertia forces an object to rotate, it will continue to do so until another force acts upon it.
Must an object be rotating to have a nonzero moment of inertia?
moment of inertia is basically an objects resistance to its state of motion.
Physical quantity coressponding to inertia is rotational motion?
The physical quantity for rotations corresponding to inertia is the moment of inertia, or rotational inertia. It is represented by the integral of r^2dm.
What is the rotational analog of mass in linear motion?
That is called moment of inertia.
What is the constant of proportionality between torque and angular acceleration?
The rotating object's moment of inertia. Similar to Newton's Second Law, commonly quoted as "force = mass x acceleration", there is an equivalent law for rotational movement: "torque = moment of inertia x angular acceleration". The moment of inertia depends on the rotating object's mass and its exact shape - you can even have a different moment of inertia for the same shape, if the axis of rotation is changed. If you use SI units, and radians for angles (and therefore radians/second2 for angular acceleration), no further constants of proportionality are required.
What physical characteristic does the moment of inertia of a rotating object most directly and accurately measure?
This is rotational inertia. When inertia forces an object to rotate, it will continue to do so until another force acts upon it.
Must an object be rotating to have a nonzero moment of inertia?
moment of inertia is basically an objects resistance to its state of motion.
Physical quantity coressponding to inertia is rotational motion?
The physical quantity for rotations corresponding to inertia is the moment of inertia, or rotational inertia. It is represented by the integral of r^2dm.
What is the property of an object that resists changes in rotational motion?
rotational inertiaMass moment if inertia.
What is the rotational analog of mass in linear motion?
That is called moment of inertia.
What increase the moment of inertia?
A rotating body that spins about an external or internal axis (either fixed or unfixed) increase the moment of inertia.
What is the constant of proportionality between torque and angular acceleration?
The rotating object's moment of inertia. Similar to Newton's Second Law, commonly quoted as "force = mass x acceleration", there is an equivalent law for rotational movement: "torque = moment of inertia x angular acceleration". The moment of inertia depends on the rotating object's mass and its exact shape - you can even have a different moment of inertia for the same shape, if the axis of rotation is changed. If you use SI units, and radians for angles (and therefore radians/second2 for angular acceleration), no further constants of proportionality are required.
Can one object have more than one rotational inertia?
YES. Infact, an object can have infinitely different moment of inertias. It all depends on the axis about which it it rotating. You can allow an object to rotate about any axis (this may or may not pass through the object).
What is rotational inertia?
Rotational inertia is sometimes called spin. It involves the movement of a mass around an axis. This moving mass will have some measure of kinetic energy that is due to the fact that it is spinning. The variables are the shape and the mass of the object, the way the mass is distributed within the object, the speed of its rotation, and the location of the axis of spin through the object. The moment of inertia might also be called angular mass, mass moment of inertia, rotational inertia, or polar moment of inertia of mass. Use the link below for more information.
Why moment of inertia is also called rotational inertia?
Because it is a measure of the "resistence" of an object to be accelerated in its rotation. An object with a big moment of inertia is more difficult to increase/decrease its angular velocity (speed of rotation), than an object with a low moment of inertia.
Why moment of inertia is called rotational inertia?
That's what it's all about: about rotation. The "inertia" part is because it is comparable to the linear inertia: that's what makes it difficult to change an object's rotation.
Why is it necessary to mention the axis of rectangular bar while calculating the moment of inertia?
An object rotating about its long axis will have a different moment of inertia than when it is rotating about its short axis. A solid disk will have a different moment than a washer, and there are formulas derived for calculating the moments of many common shapes.
