A circular disc of mass 10kg and diameter 0.5m is being rotated about an axis normal to its plane and through its centre at the rate of 1200rev/min . find angularmomentum and rotational kinetic energy about the axis of rotation

To find the angular momentum and rotational kinetic energy of the circular disc, we can use some fundamental physics formulas. Let's break down the process step by step. The given values are: mass (m) = 10 kg, diameter = 0.5 m, and the rotation rate of 1200 revolutions per minute (rev/min).

Calculating the Moment of Inertia

The moment of inertia (I) for a solid disc rotating about an axis through its center is given by the formula:

I = (1/2) * m * r²

First, we need to find the radius (r) of the disc. Since the diameter is 0.5 m, the radius will be:

r = diameter / 2 = 0.5 m / 2 = 0.25 m

Now, substituting the values into the moment of inertia formula:

I = (1/2) * 10 kg * (0.25 m)²

• I = (1/2) * 10 * 0.0625
• I = 0.3125 kg·m²

Finding Angular Velocity

Next, we need to convert the rotational speed from revolutions per minute to radians per second. The conversion is as follows:

Angular velocity (ω) = (revolutions per minute) * (2π radians/revolution) * (1 minute/60 seconds)

Plugging in the values:

ω = 1200 rev/min * (2π rad/rev) * (1 min/60 s)

• ω = 1200 * 2π / 60
• ω = 40π rad/s
• ω ≈ 125.66 rad/s

Calculating Angular Momentum

Now that we have the moment of inertia and angular velocity, we can find the angular momentum (L) using the formula:

L = I * ω

Substituting the values we found:

L = 0.3125 kg·m² * 125.66 rad/s

• L ≈ 39.25 kg·m²/s

Determining Rotational Kinetic Energy

The rotational kinetic energy (KE) can be calculated with the formula:

KE = (1/2) * I * ω²

We already have I and ω, so substituting those into the equation:

KE = (1/2) * 0.3125 kg·m² * (125.66 rad/s)²

• KE = 0.15625 * 15800.86
• KE ≈ 2470.63 Joules

Summary of Results

To summarize, for the circular disc:

• Angular Momentum (L) ≈ 39.25 kg·m²/s
• Rotational Kinetic Energy (KE) ≈ 2470.63 Joules

This exercise shows how physical properties like mass, radius, and rotation rate interact to yield important quantities in rotational dynamics. If you have further questions or need clarification on any steps, feel free to ask!