Chapter 9, Problem 1SP

Suppose the input piston of a hydraulic jack has a diameter of 3 cm and the load piston a diameter of 24 cm. The jack is being used to lift a car with a mass of 1700 kg.

1. a. What are the areas of the input and load pistons in square centimeters? (A = πr²)
2. b. What is the ratio of the area of the load piston to the area of the input piston?
3. c. What is the weight of the car in newtons? (W = mg)
4. d. What force must be applied to the input piston to support the car?

To determine

The area of input and load piston.

Answer to Problem 1SP

The area of input piston is $28.3{\text{cm}}^2$ and that of load piston is $1809.6{\text{cm}}^2$

Explanation of Solution

Given info: Diameter of input piston is $3\text{cm}$, diameter of load piston is $24\text{cm}$ mass of car is $1700\text{kg}$.

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

Substitute $3\text{cm}$ for $r$ in equation to get $A_i$.

$\begin{array}{c}{A}_i=3.14\times {\left(3\text{cm}\right)}^2\\ =28.2{\text{cm}}^2\end{array}$

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 $3\text{cm}$ for $r$ in equation to get $A_l$.

$\begin{array}{c}{A}_l=3.14\times {\left(24\text{cm}\right)}^2\\ =1809.6{\text{cm}}^2\end{array}$

Conclusion:

The area of input piston is $28.3{\text{cm}}^2$ and that of load piston is $1809.6{\text{cm}}^2$

To determine

Ratio of area of load piston to area of input piston.

Answer to Problem 1SP

The ratio is $64:1$

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 $1809.6{\text{cm}}^2$ for $A_l$ and $28.3{\text{cm}}^2$ for $A_i$ to get the ratio.

$\begin{array}{c}ratio=\frac{1809.6{\text{cm}}^2}{28.3{\text{cm}}^2}\\ =64\end{array}$

Conclusion:

The ratio is $64:1$

To determine

The weight of car.

Answer to Problem 1SP

The weight of car is $1.411\times {10}^4\text{N}$.

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 $1700\text{kg}$ for $m$ and $9.8\text{m}/{\text{s}}^{\text{2}}$ for $g$ in equation to get $W$ .

$\begin{array}{c}W=1700\text{kg}\times \text{9}\text{.8}\text{m}/{\text{s}}^{\text{2}}\\ =1.411\times {10}^4\text{N}\end{array}$

Conclusion:

The weight of car is $1.411\times {10}^4\text{N}$.

To determine

The force applied to piston to balance the car.

Answer to Problem 1SP

The force applied is $\text{498}\text{.5}\text{kPa}$.

Explanation of Solution

Write the equation to find the pressure.

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

Here,

$P$ is the pressure

$F$ is the force

$A$ is the area

Substitute $1.411\times {10}^4\text{N}$ for $F$ and $28.3{\text{cm}}^2$ for $A$ in equation to get $P$

$\begin{array}{c}P=\frac{1.411\times {10}^4\text{N}}{28.3{\text{cm}}^2}\\ =498.5\text{kPa}\end{array}$

Conclusion:

The force applied is $\text{498}\text{.5}\text{kPa}$.