Sure! Let's go through the problem step-by-step.
1. Understand the Problem:
- A machine has an efficiency of [tex]\( 70\% \)[/tex].
- It is used to raise a body of mass [tex]\( 80 \text{ kg} \)[/tex] through a vertical distance of [tex]\( 3 \text{ m} \)[/tex] in [tex]\( 40 \text{ s} \)[/tex].
- We need to calculate the power input.
- [tex]\( g \)[/tex] is given as [tex]\( 1 \text{ m/s}^2 \)[/tex].
2. Convert Efficiency into Decimal Form:
- The efficiency of the machine is given as [tex]\( 70\% \)[/tex].
- In decimal form, efficiency [tex]\( = \frac{70}{100} = 0.7 \)[/tex].
3. Calculate the Work Done:
- Work done is given by the formula: [tex]\( \text{Work} = \text{Force} \times \text{Distance} \)[/tex].
- The force in this case is the weight of the body, which is calculated as [tex]\( \text{Force} = \text{mass} \times g \)[/tex].
- Here, mass [tex]\( = 80 \text{ kg} \)[/tex] and [tex]\( g = 1 \text{ m/s}^2 \)[/tex].
- So, [tex]\( \text{Force} = 80 \text{ kg} \times 1 \text{ m/s}^2 = 80 \text{ N} \)[/tex].
- Now, the distance [tex]\( = 3 \text{ m} \)[/tex].
- Therefore, the work done [tex]\( = 80 \text{ N} \times 3 \text{ m} = 240 \text{ J} \)[/tex].
4. Calculate the Power Output:
- Power output is given by the formula: [tex]\( \text{Power} = \frac{\text{Work done}}{\text{Time}} \)[/tex].
- Here, Work done [tex]\( = 240 \text{ J} \)[/tex] and Time [tex]\( = 40 \text{ s} \)[/tex].
- So, Power output [tex]\( = \frac{240 \text{ J}}{40 \text{ s}} = 6.0 \text{ W} \)[/tex].
5. Calculate the Power Input:
- Given the efficiency, we can find the power input using the formula:
[tex]\[
\text{Efficiency} = \frac{\text{Power output}}{\text{Power input}}
\][/tex]
- Rearranging this to solve for power input:
[tex]\[
\text{Power input} = \frac{\text{Power output}}{\text{Efficiency}}
\][/tex]
- Here, Power output [tex]\( = 6.0 \text{ W} \)[/tex] and Efficiency [tex]\( = 0.7 \)[/tex].
- So, Power input [tex]\( = \frac{6.0 \text{ W}}{0.7} = 8.571428571428571 \text{ W} \)[/tex].
6. Conclusion:
- The power input required by the machine to raise the body is [tex]\( 8.57 \text{ W} \)[/tex].
