Of course! Let's break down the solution step by step:
### (i) Calculating the Work Done to Raise 1 kg of Water
Work done (W) in physics is calculated using the formula:
[tex]\[ W = m \cdot g \cdot h \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the water,
- [tex]\( g \)[/tex] is the acceleration due to gravity,
- [tex]\( h \)[/tex] is the height (or depth, in this case) the water is raised.
Given:
- Mass ([tex]\( m \)[/tex]) of the water = 1 kg
- Acceleration due to gravity ([tex]\( g \)[/tex]) = 9.8 m/s²
- Depth of the well ([tex]\( h \)[/tex]) = 200 m
Plug these values into the formula:
[tex]\[ W = 1 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 200 \, \text{m} \][/tex]
[tex]\[ W = 1960 \, \text{Joules} \][/tex]
So, the work done to raise 1 kg of water from the well is [tex]\( 1960 \, \text{Joules} \)[/tex].
### (ii) Calculating the Power Output of the Pump
Power (P) is defined as the rate at which work is done. It is calculated using the formula:
[tex]\[ P = \frac{W}{t} \][/tex]
where:
- [tex]\( W \)[/tex] is the work done,
- [tex]\( t \)[/tex] is the time taken.
However, in this case, we need to calculate the power output based on the rate of pumping water. The rate given is:
- 10 kg/min
First, we need to convert this rate from kg/min to kg/s for consistency in SI units.
[tex]\[ \text{Rate} = \frac{10 \, \text{kg}}{1 \, \text{min}} \][/tex]
Since there are 60 seconds in a minute:
[tex]\[ \text{Rate} = \frac{10 \, \text{kg}}{60 \, \text{s}} \][/tex]
[tex]\[ \text{Rate} = \frac{1}{6} \, \text{kg/s} \][/tex]
Now, we know the work done per kilogram is 1960 Joules. To find the power output:
[tex]\[ P = \text{Work done per second} \][/tex]
Since the pump lifts [tex]\( \frac{1}{6} \)[/tex] kg of water every second:
[tex]\[ P = 1960 \, \text{J} \times \frac{1}{6} \, \text{kg/s} \][/tex]
[tex]\[ P = \frac{1960}{6} \, \text{Watts} \][/tex]
[tex]\[ P \approx 326.67 \, \text{Watts} \][/tex]
So, the power output of the pump is approximately [tex]\( 326.67 \, \text{Watts} \)[/tex].
To summarize:
1. The work done to raise 1 kg of water from a 200 m deep well is [tex]\( 1960 \, \text{Joules} \)[/tex].
2. The power output of the pump, if it pumps water at a rate of 10 kg/min, is approximately [tex]\( 326.67 \, \text{Watts} \)[/tex].
