Calculate the moment of inertia of this arrangement of masses. Each mass is located 2.5 m away from the axis. (You can envision the arrangement as a single rotating axis with four masses extending out from the axis like spokes of a wheel.)

m1 is 1kg

m2 is 2kg

m3 is 3kg

m4 is 4kg

What formula for moment of inertia do I use? I was taking the sum of all values plugged into I = 4mr^2, but that was wrong. How do I find moment of inertia?

Thank you!

For a point-like mass, the moment of inertia about some axis is given by m r^2 For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in question. that does not total 4mr^2

unless you put in the 2.5 for radius.

To calculate the moment of inertia of the given arrangement of masses, you need to use the parallel axis theorem. This theorem states that the moment of inertia about an axis parallel to and at a distance "d" from an axis passing through the center of mass is given by:

I = I_cm + (M * d^2)

where I is the moment of inertia about the parallel axis, I_cm is the moment of inertia about the center of mass axis, M is the total mass of the system, and d is the distance between the two axes.

In this case, you have four masses arranged symmetrically around an axis, resembling spokes of a wheel. Since each mass is 2.5 m away from the axis, the distance "d" is 2.5 m.

Now let's calculate the moment of inertia for each mass:

For m1 (1 kg) at 2.5 m from the axis:
I1 = m1 * r1^2 = 1 kg * (2.5 m)^2 = 6.25 kg·m^2

For m2 (2 kg) at 2.5 m from the axis:
I2 = m2 * r2^2 = 2 kg * (2.5 m)^2 = 25 kg·m^2

For m3 (3 kg) at 2.5 m from the axis:
I3 = m3 * r3^2 = 3 kg * (2.5 m)^2 = 56.25 kg·m^2

For m4 (4 kg) at 2.5 m from the axis:
I4 = m4 * r4^2 = 4 kg * (2.5 m)^2 = 100 kg·m^2

Now, calculate the total moment of inertia using the parallel axis theorem:

I_total = I1 + I2 + I3 + I4 + (M * d^2)

where M is the total mass of the system, which is the sum of the individual masses (1 kg + 2 kg + 3 kg + 4 kg = 10 kg).

I_total = 6.25 kg·m^2 + 25 kg·m^2 + 56.25 kg·m^2 + 100 kg·m^2 + (10 kg * (2.5 m)^2)

Simplifying the equation further:

I_total = 6.25 kg·m^2 + 25 kg·m^2 + 56.25 kg·m^2 + 100 kg·m^2 + 62.5 kg·m^2

I_total = 250 kg·m^2

So, the moment of inertia of the given arrangement of masses is 250 kg·m^2.