Answer :
Absolutely, I'd be happy to explain the relationship between the velocity ratio (VR), the mechanical advantage (MA), and the efficiency (η) of a simple machine.
1. Velocity Ratio (VR): This is the ratio of the distance moved by the effort to the distance moved by the load in a simple machine. It essentially tells you how the input distance compares to the output distance.
2. Mechanical Advantage (MA): This is the ratio of the load force to the effort force. It indicates how much the machine amplifies the input force.
3. Efficiency (η): This is a measure of how well a simple machine converts input work into output work. It is given as a percentage.
The relationship between these three quantities can be expressed using the formula for efficiency as follows:
[tex]\[ \eta = \left( \frac{MA}{VR} \right) \times 100 \][/tex]
Here’s the detailed step-by-step reasoning:
1. Start with the basic definitions:
- [tex]\( VR = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} \)[/tex]
- [tex]\( MA = \frac{\text{Load force}}{\text{Effort force}} \)[/tex]
2. Efficiency [tex]\(\eta\)[/tex] is defined as the ratio of the useful work output to the work input, expressed as a percentage. When considering a simple machine,
[tex]\[ \eta = \left( \frac{\text{Work output}}{\text{Work input}} \right) \times 100 \][/tex]
3. Relate work output and input to forces and distances. Work done by effort is [tex]\( \text{Effort force} \times \text{Distance moved by effort} \)[/tex], and work done on the load is [tex]\( \text{Load force} \times \text{Distance moved by load} \)[/tex].
4. Substitute these into the efficiency equation:
[tex]\[ \eta = \left( \frac{\text{Load force} \times \text{Distance moved by load}}{\text{Effort force} \times \text{Distance moved by effort}} \right) \times 100 \][/tex]
5. Use the definitions of Mechanical Advantage (MA) and Velocity Ratio (VR):
- [tex]\( MA = \frac{\text{Load force}}{\text{Effort force}} \)[/tex]
- [tex]\( VR = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} \)[/tex]
6. Rearrange the efficiency formula with MA and VR:
[tex]\[ \eta = \left( \frac{\text{Load force}}{\text{Effort force}} \times \frac{\text{Distance moved by load}}{\text{Distance moved by effort}} \right) \times 100 \][/tex]
[tex]\[ \eta = \left( MA \times \frac{1}{VR} \right) \times 100 \][/tex]
[tex]\[ \eta = \left( \frac{MA}{VR} \right) \times 100 \][/tex]
So, the efficiency [tex]\(\eta\)[/tex] of a simple machine is calculated by dividing the mechanical advantage (MA) by the velocity ratio (VR) and then multiplying by 100 to express it as a percentage.
1. Velocity Ratio (VR): This is the ratio of the distance moved by the effort to the distance moved by the load in a simple machine. It essentially tells you how the input distance compares to the output distance.
2. Mechanical Advantage (MA): This is the ratio of the load force to the effort force. It indicates how much the machine amplifies the input force.
3. Efficiency (η): This is a measure of how well a simple machine converts input work into output work. It is given as a percentage.
The relationship between these three quantities can be expressed using the formula for efficiency as follows:
[tex]\[ \eta = \left( \frac{MA}{VR} \right) \times 100 \][/tex]
Here’s the detailed step-by-step reasoning:
1. Start with the basic definitions:
- [tex]\( VR = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} \)[/tex]
- [tex]\( MA = \frac{\text{Load force}}{\text{Effort force}} \)[/tex]
2. Efficiency [tex]\(\eta\)[/tex] is defined as the ratio of the useful work output to the work input, expressed as a percentage. When considering a simple machine,
[tex]\[ \eta = \left( \frac{\text{Work output}}{\text{Work input}} \right) \times 100 \][/tex]
3. Relate work output and input to forces and distances. Work done by effort is [tex]\( \text{Effort force} \times \text{Distance moved by effort} \)[/tex], and work done on the load is [tex]\( \text{Load force} \times \text{Distance moved by load} \)[/tex].
4. Substitute these into the efficiency equation:
[tex]\[ \eta = \left( \frac{\text{Load force} \times \text{Distance moved by load}}{\text{Effort force} \times \text{Distance moved by effort}} \right) \times 100 \][/tex]
5. Use the definitions of Mechanical Advantage (MA) and Velocity Ratio (VR):
- [tex]\( MA = \frac{\text{Load force}}{\text{Effort force}} \)[/tex]
- [tex]\( VR = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} \)[/tex]
6. Rearrange the efficiency formula with MA and VR:
[tex]\[ \eta = \left( \frac{\text{Load force}}{\text{Effort force}} \times \frac{\text{Distance moved by load}}{\text{Distance moved by effort}} \right) \times 100 \][/tex]
[tex]\[ \eta = \left( MA \times \frac{1}{VR} \right) \times 100 \][/tex]
[tex]\[ \eta = \left( \frac{MA}{VR} \right) \times 100 \][/tex]
So, the efficiency [tex]\(\eta\)[/tex] of a simple machine is calculated by dividing the mechanical advantage (MA) by the velocity ratio (VR) and then multiplying by 100 to express it as a percentage.
