I= 1/2 mr^2 solve for mass m.
radius given
angular momentum: I*w=I*2
KE= 1/2 I w^2= 1/2 I *2^2
mass , radius , angular momentum . . and . . kinetic energy of d disk . . . .
radius given
angular momentum: I*w=I*2
KE= 1/2 I w^2= 1/2 I *2^2
Moment of inertia, I = 75 kgm^2
Radius of gyration, k = 3 m
Angular velocity, ω = 2 rad/s
We can use the formula for moment of inertia in terms of mass and radius of the disk, I = (1/2) m R^2, where R is the radius of the disk.
(i) Solving for mass:
I = (1/2) m R^2
75 = (1/2) m R^2
m = 2I/R^2
(ii) Solving for radius:
We know that k = R/√(2), so:
3 = R/√(2)
R = 3√(2) m
(iii) Angular momentum:
Angular momentum, L = I ω
L = 75 × 2 = 150 kg m^2/s
(iv) Kinetic energy:
Kinetic energy, KE = (1/2) I ω^2
KE = (1/2) × 75 × 2^2
KE = 150 J
To find the mass of the disk, we can use the formula for moment of inertia:
I = m * r^2
where I is the moment of inertia, m is the mass, and r is the radius of gyration. Rearranging the formula, we have:
m = I / r^2
Plugging in the values, we get:
m = 75 kgm^2 / (3m)^2
m = 75 kgm^2 / 9m^2
m = 8.33 kg (approximately)
So, the mass of the disk is around 8.33 kilograms.
Now, let's bust out the formula for angular momentum:
L = I * w
where L is the angular momentum, I is the moment of inertia, and w is the angular velocity. Plugging in the values, we get:
L = 75 kgm^2 * 2 rad/s
L = 150 kgm^2/s (approximately)
So, the angular momentum of the disk is approximately 150 kilogram meter squared per second.
Finally, to find the kinetic energy of the disk, we can use the formula:
KE = (1/2) * I * w^2
where KE is the kinetic energy. Plugging in the values, we get:
KE = (1/2) * 75 kgm^2 * (2 rad/s)^2
KE = 150 J (approximately)
So, the kinetic energy of the disk is around 150 joules.
There you have it - the mass, radius, angular momentum, and kinetic energy of the disk. Hope I tickled your brain cells along the way!
1. Mass (m):
The moment of inertia (I) can be related to mass (m) and the square of the radius of gyration (k) using the formula:
I = mk^2
Rearranging the formula gives us:
m = I / k^2
Plugging in the given values, we have:
m = 75 kgm^2 / (3m)^2
m = 75 kgm^2 / 9m^2
m = 8.33 kg
So, the mass of the disk is approximately 8.33 kg.
2. Radius (r):
The radius of the disk is not directly given, but we can calculate it using the radius of gyration (k). The radius of gyration is defined as the √(I/m), where I is the moment of inertia and m is the mass.
k = √(I / m)
Plugging in the values, we get:
k = √(75 kgm^2 / 8.33 kg)
k ≈ 3 m
Therefore, the radius of the disk is approximately 3 meters.
3. Angular momentum (L):
The angular momentum (L) of a rotating object is given by the product of the moment of inertia (I) and the angular velocity (ω):
L = I * ω
Plugging in the given values, we have:
L = 75 kgm^2 * 2 rad/s
L = 150 kgm^2/s
So, the angular momentum of the disk is 150 kgm^2/s.
4. Kinetic energy (K):
The kinetic energy (K) of a rotating object can be calculated using the formula:
K = (1/2) * I * ω^2
Plugging in the given values, we get:
K = (1/2) * 75 kgm^2 * (2 rad/s)^2
K = 300 J
Therefore, the kinetic energy of the disk is 300 joules.
To summarize:
- Mass (m) ≈ 8.33 kg
- Radius (r) ≈ 3 m
- Angular momentum (L) = 150 kgm^2/s
- Kinetic energy (K) = 300 J