Each of the following objects has a radius of 0.180 m and a mass of 2.65 kg, and each rotates about an axis through its center (as in the table below) with an angular speed of 41.4 rad/s.
Find the magnitude of the angular momentum of each object.
a) a hoop ___ kg m/s
b) a solid cylinder ___ kg m/s
c) a a solid sphere ___ kg m/s
d) a hollow spherical shell ___ kg m/s
angular momentum of a hoop is m*r^2; the angular momentum of a solid cylinder is 1/2*m*r^2; the angular momentum of a solid sphere is 2/5*m*r^2; the Angular momentum of a hollow spherical shell is 2/3*m*r^2
Actually, angular momentum is L = Iw (w stands for omega and represents the angular speed). The answers listed previously are the moments of inertia (I). So, the angular momentum is equal to the moment of inertia (I) multiplied by the angular speed (w), which in this case is 41.4 rad/s. Remember that the moment of inertia (I) varies based on the shape given, meaning that there are different equations for each shape. The moment of inertia for a hoop is what the previous response says, and the same for the rest of the equations given by that person. The only thing is that those aren't equal to the angular momentum,but the moment of inertia.
a) Well, since you're a hoop, I hope you're ready for some angular momentum! The formula for the magnitude of the angular momentum of a hoop is given by L = Iω, where I is the moment of inertia and ω is the angular speed. For a hoop, the moment of inertia is given by I = mr^2, where m is the mass and r is the radius. Plugging in the values, we get L = (2.65 kg)(0.180 m)^2(41.4 rad/s). So, the magnitude of the angular momentum of the hoop is approximately 4.47 kg m/s.
b) Now for the solid cylinder, let's roll with the angular momentum formula again. The moment of inertia for a solid cylinder is given by I = (1/2)mr^2, where m is the mass and r is the radius. Plugging in the values, we get L = (1/2)(2.65 kg)(0.180 m)^2(41.4 rad/s). The magnitude of the angular momentum of the solid cylinder is approximately 1.11 kg m/s.
c) Moving on to the solid sphere, the moment of inertia is given by I = (2/5)mr^2. Plugging in the values, we get L = (2/5)(2.65 kg)(0.180 m)^2(41.4 rad/s). So, the magnitude of the angular momentum of the solid sphere is approximately 2.22 kg m/s.
d) Finally, for the hollow spherical shell, the moment of inertia is given by I = (2/3)mr^2. Plugging in the values, we get L = (2/3)(2.65 kg)(0.180 m)^2(41.4 rad/s). The magnitude of the angular momentum of the hollow spherical shell is approximately 2.96 kg m/s.
To calculate the magnitude of the angular momentum of each object, we can use the formula for angular momentum:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
a) For a hoop:
The moment of inertia of a hoop rotating about its axis is given by:
I = MR^2
Where M is the mass and R is the radius.
Substituting the given values:
M = 2.65 kg and R = 0.180 m
Calculating the moment of inertia:
I = (2.65 kg)(0.180 m)^2
Next, we need to find the angular velocity ω:
ω = 41.4 rad/s
Now, we can calculate the magnitude of the angular momentum using the formula:
L = Iω
L = (2.65 kg)(0.180 m)^2 * 41.4 rad/s
b) For a solid cylinder:
The moment of inertia of a solid cylinder rotating about its axis is given by:
I = (1/2)MR^2
Using the same values for M and R as in part a:
I = (1/2)(2.65 kg)(0.180 m)^2
Substituting the given angular velocity, we can calculate the magnitude of the angular momentum using the formula:
L = Iω
L = (1/2)(2.65 kg)(0.180 m)^2 * 41.4 rad/s
c) For a solid sphere:
The moment of inertia of a solid sphere rotating about its axis is given by:
I = (2/5)MR^2
Using the same values for M and R as before:
I = (2/5)(2.65 kg)(0.180 m)^2
Substituting the given angular velocity, we can calculate the magnitude of the angular momentum using the formula:
L = Iω
L = (2/5)(2.65 kg)(0.180 m)^2 * 41.4 rad/s
d) For a hollow spherical shell:
The moment of inertia of a hollow spherical shell rotating about its axis is given by:
I = (2/3)MR^2
Using the same values for M and R as before:
I = (2/3)(2.65 kg)(0.180 m)^2
Substituting the given angular velocity, we can calculate the magnitude of the angular momentum using the formula:
L = Iω
L = (2/3)(2.65 kg)(0.180 m)^2 * 41.4 rad/s
Now, you can substitute the values and calculate the magnitude of the angular momentum for each object.