Each of the following objects has a radius of 0.169 m and a mass of 2.02 kg, and each rotates about an axis through its center (as in this table) with an angular speed of 38.6 rad/s. Find the magnitude of the angular momentum of each object.
(a) a hoop
kg · m2/s
(b) a solid cylinder
kg · m2/s
(c) a solid sphere
kg · m2/s
(d) a hollow spherical shell
kg · m2/s
To find the magnitude of the angular momentum of each object, we can use the formula:
Angular Momentum (L) = moment of inertia (I) * angular speed (ω)
1. For a hoop:
The moment of inertia of a hoop rotating about its axis through the center is given by the formula: I = m * r^2.
So, for a hoop with a radius of 0.169 m, the moment of inertia would be:
I = 2.02 kg * (0.169 m)^2
Now, we can calculate the angular momentum:
L = I * ω
2. For a solid cylinder:
The moment of inertia of a solid cylinder rotating about its axis through the center is given by the formula: I = (1/2) * m * r^2.
So, for a solid cylinder with a radius of 0.169 m, the moment of inertia would be:
I = (1/2) * 2.02 kg * (0.169 m)^2
Now, we can calculate the angular momentum:
L = I * ω
3. For a solid sphere:
The moment of inertia of a solid sphere rotating about its axis through the center is given by the formula: I = (2/5) * m * r^2.
So, for a solid sphere with a radius of 0.169 m, the moment of inertia would be:
I = (2/5) * 2.02 kg * (0.169 m)^2
Now, we can calculate the angular momentum:
L = I * ω
4. For a hollow spherical shell:
The moment of inertia of a hollow spherical shell rotating about its axis through the center is given by the formula: I = (2/3) * m * r^2.
So, for a hollow spherical shell with a radius of 0.169 m, the moment of inertia would be:
I = (2/3) * 2.02 kg * (0.169 m)^2
Now, we can calculate the angular momentum:
L = I * ω
Using the given value of angular speed (ω) as 38.6 rad/s, substitute the corresponding values into the formulas for each object to calculate the magnitude of their angular momentum.
To find 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 speed.
(a) For a hoop, the moment of inertia is given by the formula:
I = MR²
where M is the mass of the object and R is the radius. Substituting the given values:
I = (2.02 kg)(0.169 m)² = 0.05796092 kg · m²
Once we have the moment of inertia, we can calculate the angular momentum:
L = Iω = (0.05796092 kg · m²)(38.6 rad/s) = 2.24055352 kg · m²/s
Therefore, the magnitude of the angular momentum of the hoop is 2.24055352 kg · m²/s.
(b) For a solid cylinder, the moment of inertia is given by the formula:
I = (1/2)MR²
Substituting the given values:
I = (1/2)(2.02 kg)(0.169 m)² = 0.00971944 kg · m²
Calculating the angular momentum:
L = Iω = (0.00971944 kg · m²)(38.6 rad/s) = 0.375139584 kg · m²/s
Therefore, the magnitude of the angular momentum of the solid cylinder is 0.375139584 kg · m²/s.
(c) For a solid sphere, the moment of inertia is given by the formula:
I = (2/5)MR²
Substituting the given values:
I = (2/5)(2.02 kg)(0.169 m)² = 0.01458744 kg · m²
Calculating the angular momentum:
L = Iω = (0.01458744 kg · m²)(38.6 rad/s) = 0.562634544 kg · m²/s
Therefore, the magnitude of the angular momentum of the solid sphere is 0.562634544 kg · m²/s.
(d) For a hollow spherical shell, the moment of inertia is given by the formula:
I = (2/3)MR²
Substituting the given values:
I = (2/3)(2.02 kg)(0.169 m)² = 0.01943832 kg · m²
Calculating the angular momentum:
L = Iω = (0.01943832 kg · m²)(38.6 rad/s) = 0.750278592 kg · m²/s
Therefore, the magnitude of the angular momentum of the hollow spherical shell is 0.750278592 kg · m²/s.
The final answers are:
(a) Hoop: 2.24055352 kg · m²/s
(b) Solid cylinder: 0.375139584 kg · m²/s
(c) Solid sphere: 0.562634544 kg · m²/s
(d) Hollow spherical shell: 0.750278592 kg · m²/s.