A fixed pulley is driven by a 100 kg mass falling at a rate of `8*0`m in `4*0s`. It lifts a load of `75*0` kgf. Calculate
(a) The power input to the pulley taking the force of gravity on 1 kg as 10 N.
(b) the efficiency of the pulley, and
(c) The height to which the load is raised in `4*0s`.

Answer

Let's solve the problem step by step.

### Given Data:
- Mass of falling object (effort) = 100 kg
- Rate of fall = 8 m in 4 s
- Load lifted = 75 kgf
- Gravitational force (g) = 10 N/kg

### Step 1: Calculate the Force of the Falling Mass
The force exerted by the falling mass can be calculated using the formula:
\[ \text{Force} = \text{mass} \times \text{gravity} \]
\[ \text{Force} = 100 \, \text{kg} \times 10 \, \text{N/kg} = 1000 \, \text{N} \]

### Step 2: Calculate the Power Input to the Pulley
Power input can be calculated using the formula:
\[ \text{Power} = \frac{\text{Effort} \times \text{Distance}}{\text{Time}} \]
Where:
- Effort = 1000 N (from Step 1)
- Distance = 8 m (given)
- Time = 4 s (given)

Now substituting the values:
\[ \text{Power} = \frac{1000 \, \text{N} \times 8 \, \text{m}}{4 \, \text{s}} \]
\[ \text{Power} = \frac{8000 \, \text{N m}}{4 \, \text{s}} = 2000 \, \text{W} \]

### Step 3: Calculate the Efficiency of the Pulley
Efficiency can be calculated using the formula:
\[ \text{Efficiency} = \frac{\text{Mechanical Advantage}}{\text{Velocity Ratio}} \times 100\% \]

#### Step 3.1: Calculate Mechanical Advantage (MA)
Mechanical Advantage is given by:
\[ \text{MA} = \frac{\text{Load}}{\text{Effort}} \]
Where:
- Load = 75 kgf = 75 kg × 10 N/kg = 750 N
- Effort = 1000 N (from Step 1)

Now substituting the values:
\[ \text{MA} = \frac{750 \, \text{N}}{1000 \, \text{N}} = 0.75 \]

#### Step 3.2: Calculate Velocity Ratio (VR)
For a fixed pulley, the velocity ratio is:
\[ \text{VR} = 1 \]

#### Step 3.3: Calculate Efficiency
Now substituting the values into the efficiency formula:
\[ \text{Efficiency} = \frac{0.75}{1} \times 100\% = 75\% \]

### Step 4: Calculate the Height to Which the Load is Raised
Since the velocity ratio is 1, the distance moved by the load is equal to the distance moved by the effort.
- Distance moved by effort = 8 m (given)

Thus, the height to which the load is raised is:
\[ \text{Height} = 8 \, \text{m} \]

### Final Answers:
(a) Power input to the pulley = **2000 W**
(b) Efficiency of the pulley = **75%**
(c) Height to which the load is raised = **8 m**