Particles of mass M are located at the corners of a square whose sides are of le
ID: 1965031 • Letter: P
Question
Particles of mass M are located at the corners of a square whose sides are of length D. Calculate the moment of inertia of the particles when rotated about an axis (a) through the center of the square and perpendicular to its plane (b) through the center of the square lying in the plane of the square (c) through one of the corners and perpendicular to its plane.Explanation / Answer
First, you should know that the moment of inertia of a particle of mass m rotated about a certain axis is I = m * d² , where d is the distance of this particle to this axis. You should also know that when we want to calculate the moment of inertia of a body composed by many particles, it is the sum of the moment of inertia of each particle. So, (a) We have 4 particles (one at each corner). The distance from each corner to the center is half the diagonal, (1/2)*D*sqrt(2). So we sum the 4 particles and multiply by the mass => I = 4 * M * ((1/2)*D*sqrt(2))² = 2*M*D² (b) Now you can imagine the following figure | ----|---- | | | | | | ----|---- | Ok, it's not a perfect square but try to imagine it so :) The particles are in the corner, so the distance of each particle to the axis is half the side of the square D/2. So I = 4*M*(D/2)² = M*D² (c) Now the axis is at one of the corners and perpendicular to the plane of the square 1 2 3 4 Let's suppose the axis pass through particle 1 (top left corner). - the moment of inertia of particle 1 is zero. - the moment of inertia of particles 2&3 is M*D² - the moment of inertia of particle 4 is M* (D *sqrt(2))² = 2*M*D² So, the total moment of inertia is I = 0 + 2 * M*D² + 2*M*D² = 4 MD²