To find the kinetic energy of a uniform hollow sphere rolling without slipping, we need to consider both its translational and rotational kinetic energy. Here’s how to calculate it step by step:
### Step 1: Identify the mass and speed
- Given mass \( m = 200 \, \text{g} = 0.2 \, \text{kg} \) (since \( 1 \, \text{g} = 0.001 \, \text{kg} \))
- Given speed \( v = 5.00 \, \text{cm/s} = 0.05 \, \text{m/s} \) (since \( 1 \, \text{cm} = 0.01 \, \text{m} \))
### Step 2: Calculate translational kinetic energy
The translational kinetic energy \( KE_{trans} \) is given by the formula:
\[
KE_{trans} = \frac{1}{2} m v^2
\]
Substituting the values:
\[
KE_{trans} = \frac{1}{2} \times 0.2 \, \text{kg} \times (0.05 \, \text{m/s})^2
\]
Calculating:
\[
KE_{trans} = \frac{1}{2} \times 0.2 \times 0.0025 = 0.00025 \, \text{J}
\]
### Step 3: Calculate rotational kinetic energy
For a hollow sphere, the moment of inertia \( I \) is given by:
\[
I = \frac{2}{3} m r^2
\]
However, we do not need the radius \( r \) directly because we can express the angular velocity \( \omega \) in terms of the linear velocity \( v \):
\[
\omega = \frac{v}{r}
\]
The rotational kinetic energy \( KE_{rot} \) is given by:
\[
KE_{rot} = \frac{1}{2} I \omega^2
\]
Substituting \( I \) and \( \omega \):
\[
KE_{rot} = \frac{1}{2} \left(\frac{2}{3} m r^2\right) \left(\frac{v}{r}\right)^2
\]
This simplifies to:
\[
KE_{rot} = \frac{1}{2} \left(\frac{2}{3} m r^2\right) \left(\frac{v^2}{r^2}\right) = \frac{1}{2} \cdot \frac{2}{3} m v^2 = \frac{1}{3} m v^2
\]
Substituting the values:
\[
KE_{rot} = \frac{1}{3} \times 0.2 \, \text{kg} \times (0.05 \, \text{m/s})^2
\]
Calculating:
\[
KE_{rot} = \frac{1}{3} \times 0.2 \times 0.0025 = \frac{0.00025}{3} \approx 0.0000833 \, \text{J}
\]
### Step 4: Total kinetic energy
Now, we can find the total kinetic energy \( KE \):
\[
KE = KE_{trans} + KE_{rot}
\]
Substituting the values:
\[
KE = 0.00025 \, \text{J} + 0.0000833 \, \text{J} \approx 0.0003333 \, \text{J}
\]
### Step 5: Convert to appropriate units
To express this in a more standard form:
\[
KE \approx 3.33 \times 10^{-4} \, \text{J}
\]
### Final Answer
The total kinetic energy of the hollow sphere is approximately \( 3.33 \times 10^{-4} \, \text{J} \).
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