The moment of inertia of an object depends on the shape of the object, location and orientation of the axis of rotation. For a solid disk, when the disk is rotating about an axis that is perpendicular to the disk and passing through the center of the disk, the moment of inertia is given as I = (1/2)MR2 where M is the mass of the disk and R is the radius of the disk.
(a) If the mass of the disk is 2.4 kg and the radius of the disk is 44 cm, what is the moment of inertia of the disk?
.23232 kg m2
(b) If the disk is rotating at 180 rpm, what is the angular speed of the disk in unit of rad/s? (rpm stands for "round per minute" or "revolution per minute", and is a commonly used unit for rotational speed. Thus, 1 rpm = 2 rad/min = 2/60 rad/s.)
18.85 rad/s
(c) Using the information given in the above questions (a) and (b), calculate the angular momentum of the disk.
kg m2/s
(d) Using the information given in the above questions (a) and (b), calculate the rotational kinetic energy of the disk.
J
I've answered the first two parts but I am not sure how to do last two parts can someone help?
The first two parts are correct.
I=.23232 kg-m²
ω=18.85 rad/s
3)
Angular Momentum = Iω
4)
Rotational kinetic energy = (1/2)Iω²
Sure! I can help you with the last two parts.
(c) To calculate the angular momentum of the disk, you can use the formula:
Angular momentum (L) = moment of inertia (I) * angular speed (ω)
From part (a), we know that the moment of inertia (I) is 0.23232 kg m^2. And from part (b), we know that the angular speed (ω) is 18.85 rad/s.
Therefore, the angular momentum (L) can be calculated as:
L = I * ω
L = (0.23232 kg m^2) * (18.85 rad/s)
L = 4.373632 kg m^2/s
So, the angular momentum of the disk is 4.373632 kg m^2/s.
(d) The rotational kinetic energy (K) of the disk can be calculated using the formula:
Rotational kinetic energy (K) = (1/2) * I * ω^2
We already know the moment of inertia (I) is 0.23232 kg m^2 from part (a), and the angular speed (ω) is 18.85 rad/s from part (b).
Plugging in these values, we get:
K = (1/2) * (0.23232 kg m^2) * (18.85 rad/s)^2
K = 0.5 * 0.23232 * (18.85)^2 J
K ≈ 39.264495 J
So, the rotational kinetic energy of the disk is approximately 39.26 J (rounded to two decimal places).
To find the angular momentum (L) of the disk, you can use the formula L = Iω, where I is the moment of inertia and ω is the angular speed.
In this case, you have already found the moment of inertia (I = 0.23232 kg m^2) in part (a). And in part (b), you found the angular speed (ω = 18.85 rad/s).
To find the angular momentum, you can simply multiply the moment of inertia and the angular speed:
L = I * ω
L = 0.23232 kg m^2 * 18.85 rad/s
Calculating this should give you the angular momentum of the disk.
For part (d), the formula for rotational kinetic energy (K_rot) is K_rot = (1/2)Iω^2.
You have already found the moment of inertia (I = 0.23232 kg m^2) and the angular speed (ω = 18.85 rad/s).
To find the rotational kinetic energy, substitute these values into the formula:
K_rot = (1/2)*0.23232 kg m^2 * (18.85 rad/s)^2
Calculating this will give you the rotational kinetic energy of the disk.