Rotational Inertia-
Particle 1: m=50g x=2 cm y=2 cm
P 2: m=25 x=0 y=4
P 3: m=25 x=-3 y=-3
P 4: m=30 x=-e y=4
The question says to find the rotational inertia of the collection about the x, y, and 7 axis in g/cm^2, then suppose the x axis rotational inertia is A and the y axis is B, then what is the z axis in terms of A and B?
I'm so confused on rotational inertia I don't even know where to begin
Any help would be much appreciated
To find the rotational inertia of a collection of particles, you need to sum up the individual contributions of each particle's rotational inertia. The rotational inertia of a particle about a particular axis is given by the formula:
I = m * r^2
Where:
I is the rotational inertia
m is the mass of the particle
r is the perpendicular distance from the particle to the axis of rotation
Let's calculate the rotational inertia for each particle:
Particle 1:
m = 50g = 0.05kg
x = 2cm = 0.02m
y = 2cm = 0.02m
I1_x = 0.05kg * (0.02m)^2 = 2x10^(-5) kg·m^2
I1_y = 0.05kg * (0.02m)^2 = 2x10^(-5) kg·m^2
Particle 2:
m = 25g = 0.025kg
x = 0cm = 0m
y = 4cm = 0.04m
I2_x = 0.025kg * (0)^2 = 0 kg·m^2
I2_y = 0.025kg * (0.04m)^2 = 4x10^(-5) kg·m^2
Particle 3:
m = 25g = 0.025kg
x = -3cm = -0.03m
y = -3cm = -0.03m
I3_x = 0.025kg * (-0.03m)^2 = 2.25x10^(-5) kg·m^2
I3_y = 0.025kg * (-0.03m)^2 = 2.25x10^(-5) kg·m^2
Particle 4:
m = 30g = 0.03kg
x = -e cm (e is a variable, let's assume it is a positive value) = -e cm = -e x 0.01m
y = 4cm = 0.04m
I4_x = 0.03kg * (-e x 0.01m)^2 = 0.0003e^2 kg·m^2
I4_y = 0.03kg * (0.04m)^2 = 4.8x10^(-5) kg·m^2
Now, sum up the rotational inertias for each axis:
Total rotational inertia about x-axis (A):
A = I1_x + I2_x + I3_x + I4_x
= 2x10^(-5) kg·m^2 + 0 kg·m^2 + 2.25x10^(-5) kg·m^2 + 0.0003e^2 kg·m^2
Total rotational inertia about y-axis (B):
B = I1_y + I2_y + I3_y + I4_y
= 2x10^(-5) kg·m^2 + 4x10^(-5) kg·m^2 + 2.25x10^(-5) kg·m^2 + 4.8x10^(-5) kg·m^2
Now, let's find the z-axis rotational inertia in terms of A and B:
Rotational inertia about z-axis = A + B
= (2x10^(-5) + 0 + 2.25x10^(-5) + 0.0003e^2) kg·m^2 + (2x10^(-5) + 4x10^(-5) + 2.25x10^(-5) + 4.8x10^(-5)) kg·m^2
Thus, the z-axis rotational inertia in terms of A and B is the sum of the rotational inertias about the x and y axes.