Each of the following objects has a radius of 0.174 m and a mass of 2.16 kg, and each rotates about an axis through its center (as in the table below) with an angular speed of 37.3 rad/s. Find the magnitude of the angular momentum of each object.
>Hoop of thin cylindrical shell I=MR^2
>Solid cylinder or disk I=1/2MR^2
>Long thin rod w/ rotation axis through center I= 1/12ML^2
>Solid sphere I=2/5 MR^2
>Thin spherical shell I=2/3 MR^2
>Long thin rod with rotational axis through end I=1/3ML^2
a. a hoop _______ kg*m^2/s
b. a solid cylinder ______kg*m^2/s
c. a solid sphere_________kg*m^2/s
d. a hollow spherical shell_____kg*m^2/s
To find the magnitude of the angular momentum of each object, we need to use the formula for angular momentum, which is given by:
L = I * ω
where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
Let's calculate the magnitude of the angular momentum for each object using the given values.
a. For a hoop:
The moment of inertia (I) for a hoop of thin cylindrical shell is given by I = MR^2, where M is the mass and R is the radius.
Substituting the given values, we have:
I = (2.16 kg) * (0.174 m)^2
Calculate I.
Then, substitute the value of I along with the angular speed (ω = 37.3 rad/s) into the formula for angular momentum:
L = I * ω
Calculate L. The unit of angular momentum is kg*m^2/s.
b. For a solid cylinder:
The moment of inertia (I) for a solid cylinder is given by I = 1/2MR^2.
Substitute the values of M and R into the formula. Calculate I.
Then, substitute the value of I along with the angular speed (ω = 37.3 rad/s) into the formula for angular momentum:
L = I * ω
Calculate L.
c. For a solid sphere:
The moment of inertia (I) for a solid sphere is given by I = 2/5 MR^2.
Substitute the values of M and R into the formula. Calculate I.
Then, substitute the value of I along with the angular speed (ω = 37.3 rad/s) into the formula for angular momentum:
L = I * ω
Calculate L.
d. For a hollow spherical shell:
The moment of inertia (I) for a thin spherical shell is given by I = 2/3 MR^2.
Substitute the values of M and R into the formula. Calculate I.
Then, substitute the value of I along with the angular speed (ω = 37.3 rad/s) into the formula for angular momentum:
L = I * ω
Calculate L.
By following these steps, you can find the magnitude of the angular momentum for each object.