Given data

• Initial angular speed: $600\ \text{rpm}$
• Final angular speed: $1800\ \text{rpm}$
• Time taken: $10\ \text{s}$

Convert angular speeds to rotations per second $600\ \text{rpm} = \frac{600}{60} = 10\ \text{rps}$ $1800\ \text{rpm} = \frac{1800}{60} = 30\ \text{rps}$

Use average angular speed for uniform acceleration Since the angular acceleration is uniform, the average angular speed is $\omega_{\text{avg}} = \frac{\omega_i + \omega_f}{2} = \frac{10 + 30}{2} = 20\ \text{rps}$

Find total number of rotations $N = \omega_{\text{avg}} \times t = 20 \times 10 = 200$

Final answer The number of rotations made is $\boxed{200}$