To find the moment of inertia of the system about the given axis, we need to understand the concept of the moment of inertia and how it is calculated for point masses.
The moment of inertia, denoted by I, is a measure of an object's resistance to changes in its rotational motion. It depends on the mass of the object and its distribution around the rotation axis.
For a point mass, the moment of inertia is given by the formula:
I = m * r^2
where m is the mass of the point mass and r is the perpendicular distance from the rotation axis to the point mass.
In this case, we have four point masses, each of mass 2.0 kg, placed at the corners of a square with a side length of 1.4 m. We are asked to find the moment of inertia of the system about an axis that is perpendicular to the plane of the square and passes through one of the masses.
Let's consider one of the masses at the corner of the square. The perpendicular distance from this mass to the axis of rotation is the length of the side of the square, which is 1.4 m. Therefore, the moment of inertia contributed by this mass is:
I_mass = m * r^2 = 2.0 kg * (1.4 m)^2
Since all four masses are identical and symmetrically placed, the moment of inertia contributed by each of them is the same. There are four such masses in total. Therefore, the total moment of inertia of the system can be calculated by summing up the contributions from each mass:
I_total = 4 * I_mass
Now, let's move on to the second part of the question. We are given that the system is set rotating about the axis with a kinetic energy of 216.0 J. The kinetic energy of a rotating object is given by the formula:
KE = (1/2) * I * ω^2
where KE is the kinetic energy, I is the moment of inertia, and ω is the angular velocity of the object.
Rearranging the formula, we can solve for ω:
ω = sqrt(2 * KE / I)
Plugging in the given values, we can find the angular velocity ω. However, we are tasked to find the number of revolutions per minute (rev/min). The angular velocity is measured in radians per second (rad/s). To convert from rad/s to rev/min, we need to use the conversion factor:
1 rev = 2Ï€ rad
1 min = 60 s
We can now calculate the angular velocity ω in rad/s and convert it to rev/min.
I hope this explanation helps you understand how to solve the problem step by step.