Four particles in the xy plane have the following masses and coordinates: 1 kg at (3 m, 3 m), 2 kg at (-2 m, 4 m), 3 kg at (3 m, -2 m) and 4 kg at (-3 m, -4 m). Find the total moment of inertia about:
x-axis?
y-axis?
z-axis?
9
9
4.5
To find the total moment of inertia about different axes, we need to consider the contributions from each particle.
The moment of inertia of a single particle with mass "m" about an axis "d" units away from the particle's position can be calculated using the formula: I = m * d^2.
Given the masses and coordinates of the particles, we can calculate the contribution to the moment of inertia for each particle and then sum them up to find the total moment of inertia.
Let's calculate it step by step:
1. Total Moment of Inertia about the x-axis:
To find the moment of inertia about the x-axis, we need to sum up the contributions from each particle.
a) Particle 1:
Mass = 1 kg
Distance from x-axis = 3 m (since it lies on the x-axis)
Moment of inertia of Particle 1 about the x-axis = (1 kg) * (3 m)^2 = 9 kg * m^2
b) Particle 2:
Mass = 2 kg
Distance from x-axis = 4 m (distance from (-2 m, 4 m) to x-axis)
Moment of inertia of Particle 2 about the x-axis = (2 kg) * (4 m)^2 = 32 kg * m^2
c) Particle 3:
Mass = 3 kg
Distance from x-axis = 2 m (distance from (3 m, -2 m) to x-axis)
Moment of inertia of Particle 3 about the x-axis = (3 kg) * (2 m)^2 = 12 kg * m^2
d) Particle 4:
Mass = 4 kg
Distance from x-axis = 4 m (distance from (-3 m, -4 m) to x-axis)
Moment of inertia of Particle 4 about the x-axis = (4 kg) * (4 m)^2 = 64 kg * m^2
Total moment of inertia about the x-axis = Sum of moments of inertia of all particles = 9 kg * m^2 + 32 kg * m^2 + 12 kg * m^2 + 64 kg * m^2 = 117 kg * m^2
2. Total Moment of Inertia about the y-axis:
To find the moment of inertia about the y-axis, we need to sum up the contributions from each particle.
a) Particle 1:
Mass = 1 kg
Distance from y-axis = 3 m (since it lies on the y-axis)
Moment of inertia of Particle 1 about the y-axis = (1 kg) * (3 m)^2 = 9 kg * m^2
b) Particle 2:
Mass = 2 kg
Distance from y-axis = 2 m (distance from (-2 m, 4 m) to y-axis)
Moment of inertia of Particle 2 about the y-axis = (2 kg) * (2 m)^2 = 8 kg * m^2
c) Particle 3:
Mass = 3 kg
Distance from y-axis = 5 m (distance from (3 m, -2 m) to y-axis)
Moment of inertia of Particle 3 about the y-axis = (3 kg) * (5 m)^2 = 75 kg * m^2
d) Particle 4:
Mass = 4 kg
Distance from y-axis = 4 m (distance from (-3 m, -4 m) to y-axis)
Moment of inertia of Particle 4 about the y-axis = (4 kg) * (4 m)^2 = 64 kg * m^2
Total moment of inertia about the y-axis = Sum of moments of inertia of all particles = 9 kg * m^2 + 8 kg * m^2 + 75 kg * m^2 + 64 kg * m^2 = 156 kg * m^2
3. Total Moment of Inertia about the z-axis (In the xy plane, the z-axis is perpendicular to it, so the moment of inertia about the z-axis is zero):
Since all the particles lie in the xy plane, their z-coordinates are zero, and the distances from the z-axis are also zero. Therefore, the moment of inertia about the z-axis for all the particles is zero.
So, the total moment of inertia about the z-axis is zero.
In summary:
Total moment of inertia about the x-axis = 117 kg * m^2
Total moment of inertia about the y-axis = 156 kg * m^2
Total moment of inertia about the z-axis = 0 kg * m^2
To find the total moment of inertia about the x-axis, y-axis, and z-axis, we need to calculate the moment of inertia for each particle and then sum them up.
The moment of inertia for a particle about an axis is given by the formula:
I = m * r^2
where I is the moment of inertia, m is the mass of the particle, and r is the perpendicular distance from the particle to the axis of rotation.
Let's calculate the moment of inertia for each particle and sum them up for each axis:
1. For the x-axis:
- Particle 1: I1 = m1 * y1^2 = 1 kg * (3 m)^2 = 9 kg * m^2
- Particle 2: I2 = m2 * y2^2 = 2 kg * (4 m)^2 = 32 kg * m^2
- Particle 3: I3 = m3 * y3^2 = 3 kg * (-2 m)^2 = 12 kg * m^2
- Particle 4: I4 = m4 * y4^2 = 4 kg * (-4 m)^2 = 64 kg * m^2
Total moment of inertia about the x-axis: I_total_x = I1 + I2 + I3 + I4 = 9 kg * m^2 + 32 kg * m^2 + 12 kg * m^2 + 64 kg * m^2 = 117 kg * m^2
2. For the y-axis:
- Particle 1: I1 = m1 * x1^2 = 1 kg * (3 m)^2 = 9 kg * m^2
- Particle 2: I2 = m2 * x2^2 = 2 kg * (-2 m)^2 = 8 kg * m^2
- Particle 3: I3 = m3 * x3^2 = 3 kg * (3 m)^2 = 27 kg * m^2
- Particle 4: I4 = m4 * x4^2 = 4 kg * (-3 m)^2 = 36 kg * m^2
Total moment of inertia about the y-axis: I_total_y = I1 + I2 + I3 + I4 = 9 kg * m^2 + 8 kg * m^2 + 27 kg * m^2 + 36 kg * m^2 = 80 kg * m^2
3. For the z-axis:
Since all the particles are in the xy plane, the z-coordinate for each particle is zero. Therefore, the moment of inertia about the z-axis for each particle is zero.
Total moment of inertia about the z-axis: I_total_z = 0
So, the total moment of inertia about the:
- x-axis is 117 kg * m^2
- y-axis is 80 kg * m^2
- z-axis is 0 kg * m^2