Understanding Belt Drive Power Transmission

This problem involves calculating the safe power transmitted by a belt drive system. We are given the pulley diameter, the ratio of tensions in the belt's tight and slack sides, the maximum allowable tension, and the pulley's rotational speed.

To find the power transmitted, we first need to determine the difference in tensions between the tight and slack sides of the belt, and the linear velocity of the belt.

Given Data for the Belt Drive

• Pulley Diameter (d): 200 mm = 0.2 m
• Ratio of Tensions (T₁/T₂): 1.2 (where T₁ is tight side tension, T₂ is slack side tension)
• Maximum Tension (T₁): 240 kN = 240,000 N
• Pulley Speed (N): 60 rpm

Calculating Belt Tensions

We are given the maximum tension T₁ and the ratio T₁/T₂. We can find the slack side tension T₂ using the given ratio.

$\frac{T_1}{T_2} = 1.2$

$T_2 = \frac{T_1}{1.2}$

Substituting the value of T₁:

$T_2 = \frac{240,000 \text{ N}}{1.2} = 200,000 \text{ N}$

The difference in tension (T₁ - T₂) is the effective driving force:

Driving Force = $T_1 - T_2 = 240,000 \text{ N} - 200,000 \text{ N} = 40,000 \text{ N}$

Calculating Belt Velocity

The linear velocity of the belt (v) can be calculated from the pulley's diameter (d) and rotational speed (N).

$v = \frac{\pi d N}{60}$

Substituting the given values:

$v = \frac{\pi \times 0.2 \text{ m} \times 60 \text{ rpm}}{60} = 0.2\pi \text{ m/s}$

Approximately, $v \approx 0.2 \times 3.14159 \approx 0.6283 \text{ m/s}$.

Calculating Power Transmitted

The power transmitted (P) by the belt drive is the product of the effective driving force and the belt velocity.

$P = (T_1 - T_2) \times v$

Substituting the calculated values:

$P = 40,000 \text{ N} \times 0.2\pi \text{ m/s} = 8000\pi \text{ W}$

To convert Watts (W) to kilowatts (kW), we divide by 1000:

$P = \frac{8000\pi}{1000} \text{ kW} = 8\pi \text{ kW}$

Using the approximate value of $\pi$:

$P \approx 8 \times 3.14159 \text{ kW} \approx 25.1327 \text{ kW}$

Final Result

The calculated safe power transmitted is approximately 25.13 kW. Looking at the options provided, 25 kW is the closest value.