Answer :
Certainly! Let's break down the problem step-by-step to find the forces required to move pistons [tex]$B$[/tex] and [tex]$C$[/tex] when a force of 100 N is applied on piston [tex]$A$[/tex], given the relationships between their surface areas.
### Step-by-Step Solution:
1. Identifying Relationships:
Given:
- The surface area of piston [tex]\( A \)[/tex] is 2 times that of piston [tex]\( B \)[/tex].
- The surface area of piston [tex]\( A \)[/tex] is 3 times that of piston [tex]\( C \)[/tex].
2. Defining Surface Areas:
Let's denote:
- Surface area of piston [tex]\( B \)[/tex] as [tex]\( B \)[/tex].
- Surface area of piston [tex]\( C \)[/tex] as [tex]\( C \)[/tex].
- Surface area of piston [tex]\( A \)[/tex] as [tex]\( A \)[/tex].
Using the given relationships:
- [tex]\( A = 2B \)[/tex]
- [tex]\( A = 3C \)[/tex]
3. Assuming Surface Area Values:
To simplify calculations, let's assume that the surface area of piston [tex]\( B \)[/tex] is [tex]\( 1 \)[/tex] unit. Consequently:
- [tex]\( B = 1 \)[/tex]
- [tex]\( A = 2B = 2 \times 1 = 2 \)[/tex] units.
- [tex]\( C = \frac{A}{3} = \frac{2}{3} \)[/tex] units.
4. Applying Pascal's Principle:
According to Pascal's principle:
[tex]\[ \frac{F_B}{B} = \frac{F_A}{A} \][/tex]
[tex]\[ \frac{F_C}{C} = \frac{F_A}{A} \][/tex]
Where:
- [tex]\( F_A \)[/tex] is the force applied on piston [tex]\( A \)[/tex].
- [tex]\( F_B \)[/tex] is the force on piston [tex]\( B \)[/tex].
- [tex]\( F_C \)[/tex] is the force on piston [tex]\( C \)[/tex].
Given [tex]\( F_A = 100 \)[/tex] N (force on piston [tex]\( A \)[/tex]).
5. Calculating Force on Piston B:
Using the relationship:
[tex]\[ F_B = \left( \frac{F_A}{A} \right) \times B \][/tex]
Substituting in the known values:
[tex]\[ F_B = \left( \frac{100 \text{ N}}{2 \text{ units}} \right) \times 1 \text{ unit} = 50 \text{ N} \][/tex]
6. Calculating Force on Piston C:
Using the relationship:
[tex]\[ F_C = \left( \frac{F_A}{A} \right) \times C \][/tex]
Substituting in the known values:
[tex]\[ F_C = \left( \frac{100 \text{ N}}{2 \text{ units}} \right) \times \left( \frac{2}{3} \text{ units} \right) = \frac{100 \text{ N}}{2} \times \frac{2}{3} = \frac{100 \times 2}{2 \times 3} = \frac{100}{3} \approx 33.33 \text{ N} \][/tex]
### Conclusion:
- The force required to move piston [tex]\( B \)[/tex] is [tex]\( 50 \)[/tex] N.
- The force required to move piston [tex]\( C \)[/tex] is approximately [tex]\( 33.33 \)[/tex] N.
These detailed calculations show how we can arrive at the quantities of force needed to move pistons [tex]\( B \)[/tex] and [tex]\( C \)[/tex].
### Step-by-Step Solution:
1. Identifying Relationships:
Given:
- The surface area of piston [tex]\( A \)[/tex] is 2 times that of piston [tex]\( B \)[/tex].
- The surface area of piston [tex]\( A \)[/tex] is 3 times that of piston [tex]\( C \)[/tex].
2. Defining Surface Areas:
Let's denote:
- Surface area of piston [tex]\( B \)[/tex] as [tex]\( B \)[/tex].
- Surface area of piston [tex]\( C \)[/tex] as [tex]\( C \)[/tex].
- Surface area of piston [tex]\( A \)[/tex] as [tex]\( A \)[/tex].
Using the given relationships:
- [tex]\( A = 2B \)[/tex]
- [tex]\( A = 3C \)[/tex]
3. Assuming Surface Area Values:
To simplify calculations, let's assume that the surface area of piston [tex]\( B \)[/tex] is [tex]\( 1 \)[/tex] unit. Consequently:
- [tex]\( B = 1 \)[/tex]
- [tex]\( A = 2B = 2 \times 1 = 2 \)[/tex] units.
- [tex]\( C = \frac{A}{3} = \frac{2}{3} \)[/tex] units.
4. Applying Pascal's Principle:
According to Pascal's principle:
[tex]\[ \frac{F_B}{B} = \frac{F_A}{A} \][/tex]
[tex]\[ \frac{F_C}{C} = \frac{F_A}{A} \][/tex]
Where:
- [tex]\( F_A \)[/tex] is the force applied on piston [tex]\( A \)[/tex].
- [tex]\( F_B \)[/tex] is the force on piston [tex]\( B \)[/tex].
- [tex]\( F_C \)[/tex] is the force on piston [tex]\( C \)[/tex].
Given [tex]\( F_A = 100 \)[/tex] N (force on piston [tex]\( A \)[/tex]).
5. Calculating Force on Piston B:
Using the relationship:
[tex]\[ F_B = \left( \frac{F_A}{A} \right) \times B \][/tex]
Substituting in the known values:
[tex]\[ F_B = \left( \frac{100 \text{ N}}{2 \text{ units}} \right) \times 1 \text{ unit} = 50 \text{ N} \][/tex]
6. Calculating Force on Piston C:
Using the relationship:
[tex]\[ F_C = \left( \frac{F_A}{A} \right) \times C \][/tex]
Substituting in the known values:
[tex]\[ F_C = \left( \frac{100 \text{ N}}{2 \text{ units}} \right) \times \left( \frac{2}{3} \text{ units} \right) = \frac{100 \text{ N}}{2} \times \frac{2}{3} = \frac{100 \times 2}{2 \times 3} = \frac{100}{3} \approx 33.33 \text{ N} \][/tex]
### Conclusion:
- The force required to move piston [tex]\( B \)[/tex] is [tex]\( 50 \)[/tex] N.
- The force required to move piston [tex]\( C \)[/tex] is approximately [tex]\( 33.33 \)[/tex] N.
These detailed calculations show how we can arrive at the quantities of force needed to move pistons [tex]\( B \)[/tex] and [tex]\( C \)[/tex].
