Chapter 9, Problem 1SP

(a)

To determine

The area of input and load piston.

Answer to Problem 1SP

The area of input piston is $3.14{\text{cm}}^2$ and that of load piston is $490.87{\text{cm}}^2$.

Explanation of Solution

Given info: The input piston has diameter of $2\text{ cm}$ and the output piston has diameter $25\text{ cm}$. The mass of the car is $1400\text{ kg}$.

The radius of the input piston is $1\text{ cm}$ and the radius of the output piston is $12.5\text{ cm}$.

Write the equation to find the area of input piston.

$A_i=\pi r^2$

Here,

$A_i$ is the area of input piston

$r$ is the radius of the input piston

Substitute $1\text{ cm}$ for $r$ in equation to find the area of the input piston.

$\begin{array}{c}{A}_i=\pi {\left(1\text{ cm}\right)}^2\\ =3.14{\text{cm}}^2\end{array}$

Write the equation to find the area of load piston.

$A_l=\pi r^2$

Here,

$A_l$ is the area of load piston

$r$ is the radius

Substitute $12.5\text{ cm}$ for $r$ in equation to the area of the load piston.

$\begin{array}{c}{A}_l=\pi {\left(12.5\text{ cm}\right)}^2\\ =490.87{\text{cm}}^2\end{array}$

Conclusion:

Therefore, the area of input piston is $3.14{\text{cm}}^2$ and that of load piston is $490.87{\text{cm}}^2$.

(b)

To determine

The ratio of area of load piston to area of input piston.

Answer to Problem 1SP

The ratio of area of load piston to area of input piston is $156.25:1$.

Explanation of Solution

Write the equation to find the ratio of area of load piston and input piston.

$ratio=\frac{A_l}{A_i}$

Substitute $490.87{\text{cm}}^2$ for $A_l$ and $3.14{\text{cm}}^2$ for $A_i$ to get the ratio.

$\begin{array}{c}ratio=\frac{490.87{\text{cm}}^2}{3.14{\text{cm}}^2}\\ =156.25\end{array}$

Conclusion:

Therefore, the ratio of area of load piston to area of input piston is $156.25:1$.

(c)

To determine

The weight of car in newtons.

Answer to Problem 1SP

The weight of car in newtons is $1.372\times {10}^4\text{N}$.

Explanation of Solution

Write the equation to find the weight.

$W=mg$

Here,

$W$ is the weight

$m$ is the mass

$g$ is the acceleration due to gravity

Substitute $1400\text{ kg}$ for $m$ and $9.8\text{m}/{\text{s}}^{\text{2}}$ for $g$ in equation to find the weight.$\begin{array}{c}W=\left(1400\text{ kg}\right)\left(\text{9}\text{.8}\text{m}/{\text{s}}^{\text{2}}\right)\\ =1.372\times {10}^4\text{N}\end{array}$

Conclusion:

Therefore, the weight of car in newtons is $1.372\times {10}^4\text{N}$.

(d)

To determine

The force applied to piston to support the car.

Answer to Problem 1SP

The force applied to piston to support the car is $87.8\text{N}$.

Explanation of Solution

Write the expression to find the pressure needed on the input piston to support the car of weight $1.372\times {10}^4\text{N}$.

$P=\frac{F}{A_i}$

Here,

$F$ is the force corresponding to the weight of the car

Substitute $1.372\times {10}^4\text{N}$ for $F$ and $490.87\times {10}^4{\text{m}}^2$ for $A_l$ in the above equation to find the pressure needed on the input piston to support the car of weight $1.372\times {10}^4\text{N}$.

$\begin{array}{c}P=\frac{1.372\times {10}^4\text{N}}{490.87\times {10}^4{\text{m}}^2}\\ =2.79\times {10}^{-3}\text{ Pa}\end{array}$

Write the equation to find the force applied to piston to support the car.

$F=PA$

Here,

$P$ is the pressure needed on the input piston to support the car of weight $1.372\times {10}^4\text{N}$

$F$ is the force

$A$ is the area

Substitute $2.79\times {10}^{-3}\text{ Pa}$ for $P$ and $3.14\times {10}^4{\text{m}}^2$ for $A$ in equation to find the force applied to piston to support the car.

$\begin{array}{c}F=\left(2.79\times {10}^{-3}\text{ Pa}\right)\left(3.14\times {10}^4{\text{m}}^2\right)\\ =87.76\text{N}\\ \approx 87.8\text{N}\end{array}$

Conclusion:

Therefore, the force applied to piston to support the car is $87.8\text{N}$.