Four point masses, each of mass 2.0 $kg$ are placed at the corners of a square of side 1.4 $m$. Find the moment of inertia of this system about an axis that is perpendicular to the plane of the square and passes through one of the masses.
The system is set rotating about the above axis with kinetic energy of 216.0 J. Find the number of revolutions the system makes per minute. Note: You do not need to enter the units, rev/min.
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 = 2π 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.
To find the moment of inertia of the system about the given axis, we can use the parallel axis theorem. The parallel axis theorem states that the moment of inertia of a system about an axis parallel to and a distance "d" from an axis through the center of mass is given by:
I = I_cm + Md^2
where I_cm is the moment of inertia about the center of mass and M is the total mass of the system.
Since the system is symmetric, the center of mass coincides with the center of the square. The moment of inertia of a point mass about an axis through its center of mass is given by:
I_cm = m*r^2
where m is the mass of the point mass and r is the perpendicular distance of the mass from the axis.
In this case, each point mass has a mass of 2.0 kg. The perpendicular distance of each mass from the axis passing through its center is half the side length of the square, which is 1.4 m / 2 = 0.7 m.
Therefore, the moment of inertia of each point mass about its own axis is:
I_cm = (2.0 kg) * (0.7 m)^2 = 1.96 kg·m^2
Since there are four point masses in the system, the total mass M = 4 * (2.0 kg) = 8.0 kg.
Now we can calculate the moment of inertia of the system about the given axis:
I = I_cm + Md^2 = (1.96 kg·m^2) + (8.0 kg) * (0.7 m)^2 = 8.96 kg·m^2
Next, to find the number of revolutions the system makes per minute, we can use the equation:
K.E. = (1/2)Iω^2
where K.E. is the kinetic energy of the system, I is the moment of inertia, and ω is the angular velocity.
We are given that the kinetic energy is 216.0 J.
Substituting the values into the equation, we have:
216.0 = (1/2)(8.96 kg·m^2)(ω^2)
Simplifying, we get:
ω^2 = (2*216.0)/(8.96) = 48.214
Taking the square root of both sides, we get:
ω = √(48.214) = 6.946 rad/s (rounded to the nearest thousandth)
Finally, to find the number of revolutions per minute, we can use the conversion factor 1 revolution = 2π radians. Therefore:
Number of revolutions per minute = (ω * 1 min) / (2π rad)
Substituting the values, we get:
Number of revolutions per minute = (6.946 rad/s * 60 s) / (2π rad) ≈ 66.2 rev/min
Therefore, the system makes approximately 66.2 revolutions per minute.