Estimate the power required to lift 44.5 kN load by means of a cable wrapped around a hoist. The diameter of the drum is125 cm and it rotates at 30 rpm.

To estimate the power required to lift a 44.5 kN load using a hoist with the given specifications, we need to follow these steps:

1. **Convert the drum diameter to meters**:  
   \[
   \text{Diameter} = 125 \text{ cm} = 1.25 \text{ m}
   \]
   The radius \( r \) of the drum is half of the diameter:
   \[
   r = \frac{1.25 \text{ m}}{2} = 0.625 \text{ m}
   \]

2. **Convert the rotational speed to radians per second**:  
   The rotational speed is given in revolutions per minute (rpm). To convert to radians per second (rad/s):
   \[
   \omega = 30 \text{ rpm} \times \frac{2\pi \text{ rad}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = 30 \times \frac{2\pi}{60} = \pi \text{ rad/s}
   \]

3. **Calculate the linear velocity (v) of the cable**:  
   The linear velocity is the product of the angular velocity and the radius:
   \[
   v = \omega \times r = \pi \text{ rad/s} \times 0.625 \text{ m} = 0.625\pi \text{ m/s}
   \]

4. **Convert the load to Newtons**:  
   The load is given in kilonewtons (kN). To convert to Newtons (N):
   \[
   \text{Load} = 44.5 \text{ kN} = 44,500 \text{ N}
   \]

5. **Calculate the power required**:  
   The power \( P \) required to lift the load at the given speed is the product of the force (load) and the linear velocity:
   \[
   P = F \times v = 44,500 \text{ N} \times 0.625\pi \text{ m/s}
   \]
   \[
   P \approx 44,500 \text{ N} \times 1.964 \text{ m/s} \approx 87,379 \text{ W} = 87.4 \text{ kW}
   \]

Therefore, the power required to lift the 44.5 kN load using the hoist is approximately \( 87.4 \) kW.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How is torque related to the power output of a rotating drum?
2. What is the relationship between angular velocity and linear velocity?
3. How do different drum diameters affect the power required for lifting?
4. What are the implications of different load weights on the power required?
5. How can efficiency losses in the hoist system affect the overall power calculation?

**Tip:** Always remember to convert units appropriately and consistently when solving physics problems to avoid errors.

Learn how to estimate the power required to lift a 44.5 kN load using a hoist with a drum diameter of 125 cm rotating at 30 rpm. This problem involves calculating linear velocity, converting units, and applying power formulas in mechanics.

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