Pressure
Pressure is the force applied to a surface per unit area: \[P=\frac{F}{A}.\] Pressure is a way of reformatting force (which is difficult to use with fluids) into a more applicable concept. Pressure pops up in several laws throughout fluid mechanics and thermodynamics. One of the most notable appearances is in Bernoulli's principle, which uses differences in pressure related to the speed and height of air to explain how things like airplanes and chimneys work.
Boyle's Law
Boyle's law says that the product of the pressure and the volume must remain constant if the amount of the substance present remains constant.
Boyle's Law
If the temperature and amount of an ideal gas remain constant, the product of the pressure and volume must also remain constant, or
\[P_1V_1 = P_2V_2.\]
If the pressure of a gas is doubled, its volume becomes half of the initial volume.
Charles's Law
The temperature of a system is interpreted to be the average kinetic energy of its particles. As density is the mass of a substance present in a defined volume, a higher kinetic energy of constituent particles could certainly produce a lower density.
One rule which can be used to answer problems with temperature is Charles' law, which states that the ratio of the volume to the temperature must remain constant.
Charles's Law
If the pressure of a gas is held constant, the ratio of the volume to the temperature will remain constant:
\[\frac{V_1}{T_1} = \frac{V_2}{T_2}.\]
Hydraulics
Hydraulic devices use an incompressible fluid to amplify an applied force. Since the fluid is incompressible, the pressure must be the same throughout the fluid or for two different sections: \(P_1 = P_2.\) As \(P=\frac{F}{A},\) the following theorem holds:
\[\frac{F_1}{A_1} = \frac{F_2}{A_2}.\]
Fork-lift Operator
Suppose a sleepy forklift operator parked clumsily on a hill without applying the emergency brake. Suddenly, the forklift slips and starts rolling backward down the hill. In order to stop before slamming into a schoolbus full of teddy bears, it will be necessary to apply a normal force of 150 N to the disk brake. Suppose the force is delivered by a piston of surface area 2 m\(^2\) which is connected by a hydraulic system to a foot lever of surface area 0.02 m\(^2\). With what force must the operator step on the lever in order to apply the required force? Assume there is no significant height difference from the brake to the lever.
In this problem, the fluid is transmitting the force from the operator's foot to the brake pad. Brake pads hover very close to the disc they press, so we can assume that the fluid velocity is negligible. Indeed, once the foot is clamped down on the lever, the arrangement will be static. In this case, Bernoulli's relation simplifies to
\[P_\text{foot} = P_\text{brake}.\]
As pressure is given by the force per unit area,
\[\frac{F_\text{foot}}{A_\text{foot}} = \frac{F_\text{brake}}{A_\text{brake}},\]
yielding
\[F_\text{foot} = F_\text{brake} \times \frac{A_\text{foot}}{A_\text{brake}}= 1.5 \text{ N}. \]
The asymmetry in the piston areas gives the operator a significant mechanical advantage over the braking system, making it very easy for the operator to apply the huge force required. \(_\square\)
Ideal Gas law
Main Article: ideal gas law.
The ideal gas law is the equation of the state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases. The ideal gas law is often written as follows:
Ideal Gas Law
The ideal gas law states \(PV = nRT ,\) where
- \(P\) is the pressure of the gas.
- \(V\) is the volume of the gas.
- \(n\) is the amount of substance of the gas (in moles).
- \(R\) is the ideal (or universal) gas constant, \(8.314 \frac{\text{J}}{\text{K mol}}.\)
- \(T\) is the temperature of the gas.
References
- Irfanfaiz, . Hurricane Rita Peak. Retrieved April 27, 2016, from https://commons.wikimedia.org/wiki/File:Hurricane_Rita_Peak.jpg