a uniform disk of moment of inertia 75kgm^2 and radius of gyration 3m rotates with an angular velocity of 2rad/s. determine d :

mass , radius , angular momentum . . and . . kinetic energy of d disk . . . .

A uniform disc of moment of inertia 75kgm2 and radius of gyration 3m rotates with an angular velocity of 2rad/s. determine the (i) mass (ii) radius (iii) angular momentum and (iv) kinetic energy of the disc.

Given:

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

Well, well, well, let's solve this riddle!

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!

To determine the values of mass, radius, angular momentum, and kinetic energy of the disk, we can use the given moment of inertia and radius of gyration. Here's how:

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

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