Polar Moment of Inertia: The polar moment of inertia is related to an axis which is basically perpendicular to the plane of an area. If all of the area is assumed to comprise infinitely small areas da then the polar moment of inertia is the sum of all of these areas x r 2.
The polar moment of inertia is given b. Moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. It should not be confused with the second moment of area, which is used in beam calculations.
The mass moment of inertia is often also known as the. This formula is the most 'brute force' approach to calculating the moment of inertia. The other formulas provided are usually more useful and represent the most common situations that physicists run into.
The general formula is useful if the object can be treated as a collection of discrete points which can be added up. For a more elaborate object, however, it might be necessary to apply calculus to take the integral over an entire volume. The variable r is the radius vector from the point to the axis of rotation.
The formula p r is the mass density function at each point r:. A solid sphere rotating on an axis that goes through the center of the sphere, with mass M and radius R , has a moment of inertia determined by the formula:.
A hollow sphere with a thin, negligible wall rotating on an axis that goes through the center of the sphere, with mass M and radius R , has a moment of inertia determined by the formula:. A solid cylinder rotating on an axis that goes through the center of the cylinder, with mass M and radius R , has a moment of inertia determined by the formula:. A hollow cylinder with a thin, negligible wall rotating on an axis that goes through the center of the cylinder, with mass M and radius R , has a moment of inertia determined by the formula:.
A hollow cylinder with rotating on an axis that goes through the center of the cylinder, with mass M , internal radius R 1 , and external radius R 2 , has a moment of inertia determined by the formula:.
A thin rectangular plate, rotating on an axis that's perpendicular to the center of the plate, with mass M and side lengths a and b , has a moment of inertia determined by the formula:. A thin rectangular plate, rotating on an axis along one edge of the plate, with mass M and side lengths a and b , where a is the distance perpendicular to the axis of rotation, has a moment of inertia determined by the formula:.
A slender rod rotating on an axis that goes through the center of the rod perpendicular to its length , with mass M and length L , has a moment of inertia determined by the formula:. A slender rod rotating on an axis that goes through the end of the rod perpendicular to its length , with mass M and length L , has a moment of inertia determined by the formula:.
Moment of inertia , denoted by I , measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass. For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression.
Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. This article mainly considers symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified.
Following are scalar moments of inertia. In general, the moment of inertia is a tensor, see below. A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. This expression assumes that the rod is an infinitely thin but rigid wire. This expression assumes that the shell thickness is negligible.
Skip to main content. By using our site, you agree to our collection of information through the use of cookies. To learn more, view our Privacy Policy. Log In Sign Up. Download Free PDF. Osama M Elmardi. Download PDF. A short summary of this paper. It is assumed that the student is already familiar with the following concepts. The forms of mechanical energy. All the above may be found in the pre-requisite tutorials.
The following is a summary needed for this tutorial. In engineering we normally use radians. Any angular quantity multiplied by the radius of the rotation is converted into the equivalent linear quantity as measured long the circular path. You should know that torque is a moment of force. A force applied to the axle of a wheel will not make it rotate figure 1A. A force applied at a radius will figure 1B. The torque is F r N m. Clearly it is linked with mass inertia and in fact moment of inertia means second moment of mass.
It is not only the mass that governs this reluctance but also the location of the mass. You should appreciate that a wheel with all the mass near the axle fig. Figure 2A Figure 2 B 2.
The angular velocity is Consider the case where all the mass is rotating at one radius. Figure 3 If we multiply the mass by the radius we get the first moment of mass rm If we multiply by the radius again we get the second moment of mass r2 m This second moment is commonly called the moment of inertia and has a symbol I. Unfortunately most rotating bodies do not have the mass concentrated at one radius and the moment of inertia is not calculated as easily as this.
The only problem with this approach is that the radius of gyration must be known and often this is deduced from tests on the machine. These are called elementary rings or cylinders.
Each cylinder is so thin that it may be considered as being at one radius r and cylinders. Establish the formula for the mass of one ring. The volume of the plain disc is the area of a circle radius R times the depth b.
The radius of the drum is 0. Calculate the linear and angular velocity of the wheel after it falls 0. Figure No. First calculate the change in potential Energy when it falls 0. Linear K. A cylinder has a mass of 1 kg, outer radius of 0. It is allowed to roll down an inclined plane until it has changed its height by 0. Assuming it rolls with no energy loss, calculate its linear and angular velocity at this point.
A cylinder has a mass of 3 kg, outer radius of 0. It is allowed to roll down an inclined plane until it has changed its height by 2 m. Answers Remember that the formula for the moment of nertia is as follows.
The hammer is raised to the horizontal position and allowed to swing down hitting the test sample in a vice as shown. The hammer continues in its swing to a height of 0.
Determine i. The velocity of the hammer when it strikes the sample. The energy absorbed in the impact. The angular velocity at the bottom of the swing. It is rotated by a chain drive. The sprocket on the drum has an effective diameter of 0. The force in the chain is a constant value N.
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