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?
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 add them up.
The moment of inertia for a point particle 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 distance from the particle to the axis of rotation.
Let's calculate the moment of inertia for each particle first:
Particle 1:
Mass = 1 kg
Distance from x-axis = 3 m
Distance from y-axis = 3 m
Moment of inertia about x-axis = (1 kg) * (3 m)^2 = 9 kg⋅m^2
Moment of inertia about y-axis = (1 kg) * (3 m)^2 = 9 kg⋅m^2
Particle 2:
Mass = 2 kg
Distance from x-axis = -2 m
Distance from y-axis = 4 m
Moment of inertia about x-axis = (2 kg) * (-2 m)^2 = 8 kg⋅m^2
Moment of inertia about y-axis = (2 kg) * (4 m)^2 = 32 kg⋅m^2
Particle 3:
Mass = 3 kg
Distance from x-axis = 3 m
Distance from y-axis = -2 m
Moment of inertia about x-axis = (3 kg) * (3 m)^2 = 27 kg⋅m^2
Moment of inertia about y-axis = (3 kg) * (-2 m)^2 = 12 kg⋅m^2
Particle 4:
Mass = 4 kg
Distance from x-axis = -3 m
Distance from y-axis = -4 m
Moment of inertia about x-axis = (4 kg) * (-3 m)^2 = 36 kg⋅m^2
Moment of inertia about y-axis = (4 kg) * (-4 m)^2 = 64 kg⋅m^2
Now let's sum up the moment of inertia for all particles:
Total moment of inertia about the x-axis = 9 kg⋅m^2 + 8 kg⋅m^2 + 27 kg⋅m^2 + 36 kg⋅m^2 = 80 kg⋅m^2
Total moment of inertia about the y-axis = 9 kg⋅m^2 + 32 kg⋅m^2 + 12 kg⋅m^2 + 64 kg⋅m^2 = 117 kg⋅m^2
Since all the particles are in the xy plane, the moment of inertia about the z-axis will be zero since the particles have no distance from the z-axis. Thus, the total moment of inertia about the z-axis is 0 kg⋅m^2.
So, the total moment of inertia about:
x-axis = 80 kg⋅m^2
y-axis = 117 kg⋅m^2
z-axis = 0 kg⋅m^2