Since moment of inertia is a scalar quantity, a compound object made up of several objects joined together has a moment of inertia which is the scalar sum of the individual moments of inertia.
A) A wooden cylinder has mass m and radius R. To be used as a machine part, its moment of inertia about an axis through the center and perpendicular to the flat surface must be doubled by adding a thin layer of dense material that does not significantly increase the radius. How much mass must be added to the cylinder to double I? Express the added mass in terms of the original mass m.
B) In a similar situation, a woden sphere of mass M and radius R rotates about its center. Enough mass is added uniformly over the surface to double its mass to 2M. Find the ratio of the new moment of inertia to the original value. Express as a reduced fraction.
The first paragraph is not true. Moment of inertial is about a defined axis, which is a vector quantity. About that axis, moments can be treated as a scalar.
a. moment of solid cylinder: mr^2/2
b. moment of shell cylinder: Mr^2
New cylinder is sum of both
new cylinder=2*original
Mr^2+mr^2/2= 2 mr^2
M=1.5 m
M/m=1.5 so you had to add 1.5 x as much mass to the plated outside.
A) To find the added mass required to double the moment of inertia, we need to understand the formula for the moment of inertia of a solid cylinder.
The moment of inertia of a solid cylinder about an axis perpendicular to its base and passing through its center is given by the formula:
I = (1/2) * m * R^2,
where I is the moment of inertia, m is the mass of the cylinder, and R is its radius.
To double the moment of inertia, we need to find a mass (let's call it M) that when added to the cylinder, satisfies the condition:
2 * I = (1/2) * (m + M) * R^2.
Simplifying the equation:
2 * [(1/2) * m * R^2] = (1/2) * (m + M) * R^2,
m * R^2 = (m + M) * R^2,
m = m + M,
M = 0.
From the equation, we can see that no mass needs to be added to the cylinder to double its moment of inertia. This is because the formula for moment of inertia of a cylinder is already in terms of mass and radius, and no further addition of mass is required.
B) For a solid sphere, the moment of inertia about an axis passing through its center is given by:
I = (2/5) * M * R^2,
where I is the moment of inertia, M is the mass of the sphere, and R is its radius.
To double the mass, we need to find the moment of inertia of the sphere after the additional mass is added. Let's call the new mass M'.
The new moment of inertia (I') is given by:
I' = (2/5) * (2M) * R^2.
Simplifying the equation:
I' = (4/5) * M * R^2.
Now, we can calculate the ratio of the new moment of inertia to the original moment of inertia:
I' / I = [(4/5) * M * R^2] / [(2/5) * M * R^2],
I' / I = (4/5) / (2/5),
I' / I = 4/2,
I' / I = 2.
Therefore, the ratio of the new moment of inertia to the original value is 2:1, expressed as a reduced fraction.