# Moment of Inertia of a Solid Cylinder With a Wedge Removed

• cedoty1989
In summary, the moment of inertia of a solid cylinder with a wedge removed is a measure of its resistance to changes in angular acceleration. It can be calculated by multiplying the mass of the cylinder by the square of its radius and the fraction of its total volume that is removed by the wedge. To calculate it, you will need to know the mass, radius, and volume of the cylinder and the angle and volume of the wedge. This value is important in physics and engineering as it helps determine an object's behavior under rotational forces and can be used to calculate its kinetic energy and angular velocity. Removing a wedge from the cylinder will decrease its moment of inertia, as it reduces the mass and volume of the cylinder. The moment of inertia can also be
cedoty1989
Homework Statement
Imagine a solid cylinder height h able to rotate vertically around z-axis (centered at x=0 y=0). There is a wedge cut out so that when looking down with z hat pointing out of the page there is an angle 2a formed such that their is symmetry with respect to reflection over the x axis (See picture). The question is to calculate the moment of inertia.
Relevant Equations
I=∫∫∫ dm r^2 -> Cylindrical Coords: (r from 0 to R) (z from zero to h) (theta from -Pi + a to Pi-a)

Uniform density -> dm=(M/V)dV

dV = (dr)r(dtheta)(dz)

The rest of the equations are in the picture...

#### Attachments

• CamScanner 11-23-2020 21.50-1.jpg
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I know there should be an alpha in there somewhere but I cannot see where I'm going wrong. Thank you in advance.

cedoty1989 said:
The question is to calculate the moment of inertia.
About what axis? z axis through origin or through mass centre?
Your r22+z2 doesn't make sense for either.

## 1. What is the formula for calculating the moment of inertia of a solid cylinder with a wedge removed?

The formula for calculating the moment of inertia of a solid cylinder with a wedge removed is I = (1/2)MR^2, where M is the mass of the cylinder and R is the radius of the cylinder.

## 2. How does the removal of a wedge affect the moment of inertia of a solid cylinder?

The removal of a wedge from a solid cylinder decreases the moment of inertia because it reduces the mass and distance from the axis of rotation, resulting in a smaller value for the moment of inertia.

## 3. What is the significance of the moment of inertia in physics?

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is an important concept in physics, as it is used to calculate the angular acceleration of an object and understand its behavior in rotational motion.

## 4. How does the moment of inertia of a solid cylinder with a wedge removed compare to that of a solid cylinder without a wedge removed?

The moment of inertia of a solid cylinder with a wedge removed is smaller than that of a solid cylinder without a wedge removed. This is because the removal of the wedge reduces the mass and distance from the axis of rotation, resulting in a lower moment of inertia value.

## 5. Can the moment of inertia of a solid cylinder with a wedge removed be negative?

No, the moment of inertia cannot be negative. It is always a positive value, as it represents the rotational inertia of an object and cannot be in the opposite direction of the rotational motion.

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