Four particles in the xy plane have the following masses and coordinates: 1 kg at (4 m, 4 m), 2 kg at (−3 m, 5 m), 3 kg at (5 m, −2 m) and 4 kg at (−2 m, −4 m). Find the total moment of inertia about:

a) the x-axis:
kg*m2

b) the y-axis:
kg*m2

b) the z-axis:
kg*m2

a) Add M*y^2 for the four particles

b) Add M*x^2 for the four particles
c) Add M(x^2 + y^2) for the four particles. (Or just add the first two answers)

a) Well, to find the total moment of inertia about the x-axis, we need to consider the masses and distances of the particles from the x-axis. Since the x-axis is a one-dimensional line, we can treat it as the z-axis for simplicity.

Let's calculate the moment of inertia for each particle and then sum them up:

For the 1 kg particle at (4 m, 4 m):
Mass = 1 kg
Distance from the x-axis (z-axis) = 4 m (since the x-axis is also the z-axis in this case)
Moment of inertia = Mass * Distance squared = 1 kg * (4 m)^2 = 16 kg*m^2

For the 2 kg particle at (-3 m, 5 m):
Mass = 2 kg
Distance from the x-axis (z-axis) = 5 m
Moment of inertia = Mass * Distance squared = 2 kg * (5 m)^2 = 50 kg*m^2

For the 3 kg particle at (5 m, -2 m):
Mass = 3 kg
Distance from the x-axis (z-axis) = 2 m
Moment of inertia = Mass * Distance squared = 3 kg * (2 m)^2 = 12 kg*m^2

For the 4 kg particle at (-2 m, -4 m):
Mass = 4 kg
Distance from the x-axis (z-axis) = 4 m
Moment of inertia = Mass * Distance squared = 4 kg * (4 m)^2 = 64 kg*m^2

Now, let's add up the moments of inertia for all the particles:

Total moment of inertia about the x-axis = 16 kg*m^2 + 50 kg*m^2 + 12 kg*m^2 + 64 kg*m^2 = 142 kg*m^2

So, the total moment of inertia about the x-axis is 142 kg*m^2.

b) Similarly, to find the total moment of inertia about the y-axis, we need to consider the masses and distances of the particles from the y-axis. Since the y-axis is also a one-dimensional line, we can also treat it as the z-axis.

Let's calculate the moment of inertia for each particle and then sum them up:

For the 1 kg particle at (4 m, 4 m):
Mass = 1 kg
Distance from the y-axis (z-axis) = 4 m (since the y-axis is also the z-axis in this case)
Moment of inertia = Mass * Distance squared = 1 kg * (4 m)^2 = 16 kg*m^2

For the 2 kg particle at (-3 m, 5 m):
Mass = 2 kg
Distance from the y-axis (z-axis) = 3 m
Moment of inertia = Mass * Distance squared = 2 kg * (3 m)^2 = 18 kg*m^2

For the 3 kg particle at (5 m, -2 m):
Mass = 3 kg
Distance from the y-axis (z-axis) = 5 m
Moment of inertia = Mass * Distance squared = 3 kg * (5 m)^2 = 75 kg*m^2

For the 4 kg particle at (-2 m, -4 m):
Mass = 4 kg
Distance from the y-axis (z-axis) = 4 m
Moment of inertia = Mass * Distance squared = 4 kg * (4 m)^2 = 64 kg*m^2

Now, let's add up the moments of inertia for all the particles:

Total moment of inertia about the y-axis = 16 kg*m^2 + 18 kg*m^2 + 75 kg*m^2 + 64 kg*m^2 = 173 kg*m^2

So, the total moment of inertia about the y-axis is 173 kg*m^2.

c) Unfortunately, we can only find the moment of inertia about the z-axis for particles that are not confined to a 2-dimensional plane. Since all the particles are in the xy plane, the moment of inertia about the z-axis is not applicable in this scenario.

Sorry, no moment of inertia about the z-axis this time!

To find the total moment of inertia about an axis, we need to determine the moment of inertia for each particle and then sum them up.

a) For the x-axis:

The moment of inertia for a particle about the x-axis is given by the expression: Ix = m*y^2, where m is the mass of the particle and y is the perpendicular distance between the particle and the x-axis.

For the 1 kg particle at (4 m, 4 m), the perpendicular distance from the x-axis is 4 m. Therefore, I1x = 1 kg * (4 m)^2 = 16 kg*m^2.

For the 2 kg particle at (-3 m, 5 m), the perpendicular distance from the x-axis is 5 m. Therefore, I2x = 2 kg * (5 m)^2 = 50 kg*m^2.

For the 3 kg particle at (5 m, -2 m), the perpendicular distance from the x-axis is 2 m. Therefore, I3x = 3 kg * (2 m)^2 = 12 kg*m^2.

For the 4 kg particle at (-2 m, -4 m), the perpendicular distance from the x-axis is 4 m. Therefore, I4x = 4 kg * (4 m)^2 = 64 kg*m^2.

The total moment of inertia about the x-axis is the sum of the individual moment of inertia values: Itotalx = I1x + I2x + I3x + I4x = 16 kg*m^2 + 50 kg*m^2 + 12 kg*m^2 + 64 kg*m^2 = 142 kg*m^2.

Therefore, the total moment of inertia about the x-axis is 142 kg*m^2.

b) For the y-axis:

The moment of inertia for a particle about the y-axis is given by the expression: Iy = m*x^2, where m is the mass of the particle and x is the perpendicular distance between the particle and the y-axis.

Using the same coordinates and masses as in part a, we can calculate the individual moment of inertia values.

I1y = 1 kg * (4 m)^2 = 16 kg*m^2.

I2y = 2 kg * (-3 m)^2 = 18 kg*m^2.

I3y = 3 kg * (5 m)^2 = 75 kg*m^2.

I4y = 4 kg * (-2 m)^2 = 16 kg*m^2.

The total moment of inertia about the y-axis is the sum of the individual moment of inertia values: Itotaly = I1y + I2y + I3y + I4y = 16 kg*m^2 + 18 kg*m^2 + 75 kg*m^2 + 16 kg*m^2 = 125 kg*m^2.

Therefore, the total moment of inertia about the y-axis is 125 kg*m^2.

c) For the z-axis:

Since all the particles are in the xy plane, the perpendicular distance from the z-axis for all particles is zero. Therefore, the moment of inertia about the z-axis for each particle is zero.

The total moment of inertia about the z-axis is the sum of the individual moment of inertia values, which are all zero: Itotalz = I1z + I2z + I3z + I4z = 0 + 0 + 0 + 0 = 0.

Therefore, the total moment of inertia about the z-axis is 0 kg*m^2.

To find the total moment of inertia about a given axis, we need to consider the mass and the distance of each particle from that axis.

a) Total moment of inertia about the x-axis:
To find the moment of inertia about the x-axis, we need to consider the y-coordinate of each particle.

Given:
Particle 1: 1 kg at (4 m, 4 m)
Particle 2: 2 kg at (−3 m, 5 m)
Particle 3: 3 kg at (5 m, −2 m)
Particle 4: 4 kg at (−2 m, −4 m)

The formula to calculate the moment of inertia for each particle about the x-axis is: Ix = m * y^2, where m is the mass and y is the distance from the x-axis.

For Particle 1:
Ix1 = 1 kg * (4 m)^2 = 16 kg*m^2

For Particle 2:
Ix2 = 2 kg * (5 m)^2 = 50 kg*m^2

For Particle 3:
Ix3 = 3 kg * (−2 m)^2 = 12 kg*m^2

For Particle 4:
Ix4 = 4 kg * (−4 m)^2 = 64 kg*m^2

To find the total moment of inertia about the x-axis, we sum up the individual moments of inertia of each particle:
Total moment of inertia about the x-axis (Ix) = Ix1 + Ix2 + Ix3 + Ix4 = 16 kg*m^2 + 50 kg*m^2 + 12 kg*m^2 + 64 kg*m^2 = 142 kg*m^2

b) Total moment of inertia about the y-axis:
To find the moment of inertia about the y-axis, we need to consider the x-coordinate of each particle.

The formula to calculate the moment of inertia for each particle about the y-axis is: Iy = m * x^2, where m is the mass and x is the distance from the y-axis.

For Particle 1:
Iy1 = 1 kg * (4 m)^2 = 16 kg*m^2

For Particle 2:
Iy2 = 2 kg * (−3 m)^2 = 18 kg*m^2

For Particle 3:
Iy3 = 3 kg * (5 m)^2 = 75 kg*m^2

For Particle 4:
Iy4 = 4 kg * (−2 m)^2 = 16 kg*m^2

To find the total moment of inertia about the y-axis, we sum up the individual moments of inertia of each particle:
Total moment of inertia about the y-axis (Iy) = Iy1 + Iy2 + Iy3 + Iy4 = 16 kg*m^2 + 18 kg*m^2 + 75 kg*m^2 + 16 kg*m^2 = 125 kg*m^2

c) Total moment of inertia about the z-axis:
In the given problem, all the particles lie in the xy plane, so they do not contribute to the moment of inertia about the z-axis. Therefore, the total moment of inertia about the z-axis is zero (0 kg*m^2).

Therefore, the answers are:

a) Total moment of inertia about the x-axis: 142 kg*m^2
b) Total moment of inertia about the y-axis: 125 kg*m^2
c) Total moment of inertia about the z-axis: 0 kg*m^2