To solve the problem of finding the total kinetic energy of a disc rolling without slipping, we can break down the solution into several steps:
### Step 1: Understand the Problem
We have a disc with a mass \( m = 4.8 \, \text{kg} \) and radius \( r = 1 \, \text{m} \) rolling on a horizontal surface without slipping. The angular velocity is given as \( 600 \, \text{rotations/min} \).
### Step 2: Convert Angular Velocity to Radians per Second
First, we need to convert the angular velocity from rotations per minute to radians per second.
\[
\text{Angular velocity} (\omega) = 600 \, \text{rotations/min} \times \frac{2\pi \, \text{radians}}{1 \, \text{rotation}} \times \frac{1 \, \text{min}}{60 \, \text{s}}
\]
Calculating this gives:
\[
\omega = 600 \times \frac{2\pi}{60} = 10 \times 2\pi = 20\pi \, \text{radians/s}
\]
### Step 3: Determine the Linear Velocity
Since the disc is rolling without slipping, the linear velocity \( V \) can be calculated using the relationship:
\[
V = \omega \cdot r
\]
Substituting the values:
\[
V = 20\pi \cdot 1 = 20\pi \, \text{m/s}
\]
### Step 4: Calculate the Translational Kinetic Energy
The translational kinetic energy (\( KE_{\text{trans}} \)) is given by the formula:
\[
KE_{\text{trans}} = \frac{1}{2} m V^2
\]
Substituting the values:
\[
KE_{\text{trans}} = \frac{1}{2} \cdot 4.8 \cdot (20\pi)^2
\]
Calculating \( (20\pi)^2 \):
\[
(20\pi)^2 = 400\pi^2
\]
Thus,
\[
KE_{\text{trans}} = \frac{1}{2} \cdot 4.8 \cdot 400\pi^2 = 960\pi^2 \, \text{J}
\]
### Step 5: Calculate the Rotational Kinetic Energy
The rotational kinetic energy (\( KE_{\text{rot}} \)) for a disc is given by:
\[
KE_{\text{rot}} = \frac{1}{2} I \omega^2
\]
For a disc, the moment of inertia \( I \) is:
\[
I = \frac{1}{2} m r^2
\]
Substituting the values:
\[
I = \frac{1}{2} \cdot 4.8 \cdot (1)^2 = 2.4 \, \text{kg m}^2
\]
Now substituting \( I \) and \( \omega \):
\[
KE_{\text{rot}} = \frac{1}{2} \cdot 2.4 \cdot (20\pi)^2
\]
Calculating this:
\[
KE_{\text{rot}} = \frac{1}{2} \cdot 2.4 \cdot 400\pi^2 = 480\pi^2 \, \text{J}
\]
### Step 6: Calculate Total Kinetic Energy
The total kinetic energy (\( KE_{\text{total}} \)) is the sum of the translational and rotational kinetic energies:
\[
KE_{\text{total}} = KE_{\text{trans}} + KE_{\text{rot}} = 960\pi^2 + 480\pi^2 = 1440\pi^2 \, \text{J}
\]
### Final Answer
The total kinetic energy of the disc is:
\[
\boxed{1440\pi^2 \, \text{J}}
\]
