The figure below

shows an arrangement of 15 identical disks that have been glued together in a rod-like shape of length L = 1.4500 m and (total) mass M = 135.000 g. The arrangement can rotate about a perpendicular axis through its central disk at point O.
(a) What is the rotational inertia of the arrangement about that axis?

(b) If we approximated the arrangement as being a uniform rod of mass M and length L, what percentage error would we make in using the formula in Table 10-2e to calculate the rotational inertia?
%

the formula given is I=1/12 ML^2

(a) To find the rotational inertia of the arrangement about the given axis, we need to use the parallel-axis theorem. This theorem states that the rotational inertia of an object about an axis parallel to and a distance 'd' away from an axis through its center of mass is given by the sum of its rotational inertia about the center of mass (I_cm) and the product of its total mass (M) and the square of the distance 'd'. Let's calculate it step by step:

Step 1: Find the rotational inertia of one disk about the given axis.
Since all the disks are identical, we can calculate the rotational inertia of one disk (I_disk) and then multiply it by the number of disks (N = 15) to get the total rotational inertia.

The formula for the rotational inertia of a solid disk rotating about an axis perpendicular to its plane and passing through its center is given by:
I_disk = (1/2) * M_disk * R^2

where M_disk is the mass of one disk and R is its radius.

Step 2: Find the rotational inertia of one disk about its own axis through its center.
To apply the parallel-axis theorem, we need to find the rotational inertia of one disk about its own axis through its center, which can be calculated using the formula:
I_self = (1/2) * M_disk * (2R)^2

Step 3: Find the distance d between the given axis and the axis through the center of mass.
Since the arrangement is symmetrical and all the disks have the same radius, the center of mass of the arrangement coincides with the center of the middle disk. Therefore, the distance between the given axis and the axis through the center of mass is equal to half the length of the rod.

d = L/2

Step 4: Apply the parallel-axis theorem.
Using the parallel-axis theorem, we can calculate the rotational inertia of one disk about the given axis:
I_disk_given_axis = I_self + M_disk * d^2

Step 5: Calculate the total rotational inertia of the arrangement.
The total rotational inertia of the arrangement (I_arrangement) is obtained by multiplying the rotational inertia of one disk about the given axis by the number of disks:
I_arrangement = N * I_disk_given_axis

(b) To calculate the percentage error in using the formula in Table 10-2e (I_uniform_rod = (1/12) * M * L^2) as an approximation, we need to compare it with the actual value of rotational inertia (I_arrangement) calculated in part (a).
The percentage error can be calculated using the formula:
% error = [(I_uniform_rod - I_arrangement) / I_arrangement] * 100%

Substituting the values into the formulas, we can calculate the answers.

To calculate the rotational inertia of the disk arrangement, we need to consider the rotational inertia of each individual disk and then sum them up.

The rotational inertia of an individual disk rotating about its center can be calculated using the formula:

I_disk = (1/2) * M_disk * R_disk^2

where M_disk is the mass of each disk and R_disk is the radius of each disk.

Since all the disks in the arrangement are identical, their rotational inertia will be the same. Let's calculate the rotational inertia of one disk and then multiply it by the number of disks to get the total rotational inertia of the arrangement.

Given information:
L = 1.4500 m (length of the rod-like shape)
M = 135.000 g (total mass of the arrangement)
Formula: I = (1/2) * M * R^2

First, let's convert the total mass from grams to kilograms:
M = 135.000 g = 0.135 kg

To calculate the rotational inertia, we need to find the radius of each disk.
Since the disks are glued together in a rod-like shape, we can assume that the radius of each disk is the same as the thickness of the rod. However, the thickness is not given.

For part (a), to calculate the rotational inertia accurately, we need to know the radius or thickness of the disks. Without this information, it is not possible to calculate the rotational inertia.

For part (b), let's assume that we can approximate the arrangement as a uniform rod with mass M and length L.

The formula for the rotational inertia of a uniform rod rotating about its center is:

I_rod = (1/12) * M_rod * L^2

To calculate the percentage error in using the formula for the uniform rod, we can compare the rotational inertia of the disk arrangement (which we cannot calculate without the disk thickness) to the rotational inertia of the uniform rod.

To find the percentage error, we need to calculate the difference between the two values and divide it by the rotational inertia of the disk arrangement. Then, multiply by 100 to express it as a percentage.

Error percentage = (I_rod - I_disk) / I_disk * 100%

Please provide the value of the uniform rod mass M_rod so we can calculate the percentage error.

I don't quite get the picture here. The exact moment of inertia, for rotation about an axis that is perpendicular to the axis of the glued-together rod, will depend upon the diameter of the discs as well as the length of the rod. I don't get the significance of the number of discs, since they are glued together into one road.

The formula I = (1/2) m L^2 applies to a rod that is very narrow compared to its length, and rotated about its center, about an axis PERPENDICULAR to the axis of the rod

Use parallel axis theorem for each individual particle