Sure! Let's solve this step-by-step.
### Step 1: Calculate the Force
First, we need to calculate the force exerted by the crate. The force is given by the formula:
[tex]\[ \text{Force} = \text{mass} \times g \][/tex]
where
- mass is the mass of the crate, which is 500 kg.
- [tex]\( g \)[/tex] is the acceleration due to gravity, which is [tex]\( 10 \, \text{m/s}^2 \)[/tex].
So,
[tex]\[ \text{Force} = 500 \, \text{kg} \times 10 \, \text{m/s}^2 = 5000 \, \text{N} \][/tex]
### Step 2: Calculate the Work Done
Next, we calculate the work done by the machine to lift the crate. The work done is given by the formula:
[tex]\[ \text{Work} = \text{Force} \times \text{distance} \][/tex]
where
- the distance is the height through which the crate is lifted, which is 20 meters.
So,
[tex]\[ \text{Work} = 5000 \, \text{N} \times 20 \, \text{m} = 100000 \, \text{J} \][/tex]
### Step 3: Calculate the Power Output
We then calculate the power output of the machine. Power is given by:
[tex]\[ \text{Power} = \frac{\text{Work}}{\text{time}} \][/tex]
where
- the time taken is 60 seconds.
So,
[tex]\[ \text{Power} = \frac{100000 \, \text{J}}{60 \, \text{s}} = 1666.67 \, \text{W} \][/tex]
### Step 4: Convert Power Output to Kilowatts
Finally, we convert the power output from watts to kilowatts. Since 1 kilowatt (kW) is equal to 1000 watts (W), we get:
[tex]\[ \text{Power in kilowatts} = \frac{1666.67 \, \text{W}}{1000} = 1.67 \, \text{kW} \][/tex]
Therefore, the power output of the machine is [tex]\( 1.67 \, \text{kW} \)[/tex].
