Chapter 17

Boyle's law states that for a confined gas at a constant temperature, the product of the pressure and the volume is a constant. Another way of stating this law is that the pressure is inversely proportional to the volume, or that the volume is inversely proportional to the pressure. Assume a constant temperature in the following problems. A certain quantity of gas, when compressed to a volume of \(2.50 \mathrm{m}^{3},\) has a pressure of 184 Pa. The pascal (Pa) is the SI unit of pressure. It equals 1 newton per square meter. Find the pressure resulting when that gas is further compressed to \(1.60 \mathrm{m}^{3}\).

A U.S. Geological Survey map has a scale of \(1: 24,000 .\) If two buildings are 3.86 miles apart, how many inches apart are they on the map?

The resistance of a conductor is directly proportional to its length. If the resistance of \(2.60 \mathrm{mi}\) of a certain transmission line is \(155 \Omega,\) find the resistance of \(75.0 \mathrm{mi}\) of that line.

When an electric current flows through a wire, the resistance to the flow varies directly as the length and inversely as the cross-sectional area of the wire. If the length and the diameter are both tripled, by what factor will the resistance change?

Graph the power function \(y=0.553 x^{5}\) for \(x=-3\) to 3

Boyle's law states that for a confined gas at a constant temperature, the product of the pressure and the volume is a constant. Another way of stating this law is that the pressure is inversely proportional to the volume, or that the volume is inversely proportional to the pressure. Assume a constant temperature in the following problems. The air in a cylinder is at a pressure of \(14.7 \mathrm{lb} / \mathrm{in.}^{2}\) and occupies a volume of 175 in. \(^{3}\) Find the pressure when it is compressed to 25.0 in. \(^{3} .\)

Insert the missing quantity. $$\frac{x+2}{5 x}=\frac{?}{5}$$

The resistance of a certain spool of wire is \(1120 \Omega\). A piece 10.0 m long is found to have a resistance of \(12.3 \Omega .\) Find the length of wire on the spool.

If \(750 \mathrm{m}\) of \(3.00-\mathrm{mm}\) -diameter wire has a resistance of \(27.6 \Omega,\) what length of similar wire \(5.00 \mathrm{mm}\) in diameter will have the same resistance?

Graph the power function \(y=1.25 x^{3 / 2}\) for \(x=0\) to 5

Boyle's law states that for a confined gas at a constant temperature, the product of the pressure and the volume is a constant. Another way of stating this law is that the pressure is inversely proportional to the volume, or that the volume is inversely proportional to the pressure. Assume a constant temperature in the following problems. A balloon contains \(320 \mathrm{m}^{3}\) of gas at a pressure of \(140,000 \mathrm{Pa}\). What would the volume be if the same quantity of gas were at a pressure of \(250,000 \mathrm{Pa} ?\)

Find the mean proportional between the following. 2 and 50

If a certain machine can make 1850 parts in 55 min, how many parts can it make in 7.5 h? Work to the nearest part.

Newton's law of gravitation states that every body in the universe attracts every other body with a force that varies directly as the product of their masses and inversely as the square of the distance between them. By what factor will the force change when the distance is doubled and each mass is tripled?

For these problems, we use Eq. \(1018, s=v_{0} t+a t^{2} / 2\) which says that, if assume the initial velocity is zero, the distance \(s\) fallen by a body (from rest) varies directly as the square of the elapsed time \(t\) If a body falls \(176 \mathrm{m}\) in \(6.00 \mathrm{s}\), how far will it fall in \(9.00 \mathrm{s} ?\)

Newton's law of gravitation states that any two bodies attract each other with a force that is inversely proportional to the square of the distance between them. The force of attraction between two certain steel spheres is \(3.75 \times 10^{-5}\) dyne when the spheres are placed \(18.0 \mathrm{cm}\) apart. Find the force of attraction when they are \(52.0 \mathrm{cm}\) apart.

Find the mean proportional between the following. 3 and 48

In Fig. \(17-16,\) the constant force on the plunger keeps the pressure of the gas in the cylinder constant. The piston rises when the gas is heated and falls when the gas is cooled. If the volume of the gas is \(1520 \mathrm{cm}^{3}\) when the temperature is \(302 \mathrm{K},\) find the volume when the temperature is \(358 \mathrm{K}\).

If both masses are increased by \(60 \%\) and the distance between them is halved, by what percent will the force of attraction increase?

A scale model of a space capsule has a scale of \(1: 8 .\) Its volume is found, by immersion in water, to be \(556,000 \mathrm{cm}^{3} .\) Find the volume of the actual capsule, in cubic meters.

Find the mean proportional between the following. 6 and 150

The power generated by a hydroelectric plant is directly proportional to the flow rate through the turbines, and a flow rate of 5625 gallons of water per minute produces \(41.2 \mathrm{MW}\). How much power would you expect when a drought reduces the flow to 5000 gal/min?

The intensity of illumination at a given point is directly proportional to the intensity of the light source and inversely proportional to the square of the distance from the light source. If a desk is properly illuminated by a 75.0 - \(\mathrm{W}\) lamp 8.00 ft from the desk, what size lamp will be needed to provide the same lighting at a distance of \(12.0 \mathrm{ft} ?\)

Newton's law of gravitation states that any two bodies attract each other with a force that is inversely proportional to the square of the distance between them. The force of attraction between the earth and some object is called the weight of that object. The law of gravitation states, then, that the weight of an object is inversely proportional to the square of its distance from the center of the earth. If a person weighs 150 lb on the surface of the earth (assume this to be 3960 mi from the center), how much will he weigh 1500 mi above the surface of the earth?

Writing: Suppose that your company makes plastic trays and is planning new ones with dimensions, including thickness of the material, double those now being made. Your company president is convinced that they will need only twice as much plastic as the older version. "Twice the size, twice the plastic," he proclaims, and no one is willing to challenge him. Your job is to make a presentation to the president where you tactfully point out that he is wrong and where you explain that the new trays will require eight times as much plastic. Write your presentation.

Find the mean proportional between the following. 5 and 45

For these problems, we use Eq. \(1018, s=v_{0} t+a t^{2} / 2\) which says that, if assume the initial velocity is zero, the distance \(s\) fallen by a body (from rest) varies directly as the square of the elapsed time \(t\) If a body falls 738 units in \(3.00 \mathrm{s}\), how far will it fall in \(6.00 \mathrm{s} ?\)

Find the mean proportional between the following. 4 and 36

The volume of a given weight of gas varies directly as its absolute temperature \(t\) and inversely as its pressure \(p .\) If the volume is \(4.45 \mathrm{m}^{3}\) when \(p=225\) kilopascals (kPa) and \(t=305 \mathrm{K},\) find the volume when \(p=325 \mathrm{kPa}\) and \(t=354 \mathrm{K}\).

From Eq. \(1066, P=I^{2} R,\) we see that power \(P\) dissipated in a resistor varies directly as the square of the current \(I\) in the resistor. If the power dissipated in a resistor is \(486 \mathrm{W}\) when the current is \(2.75 \mathrm{A}\), find the power when the current is \(3.45 \mathrm{A}\)

The inverse square law states that for a surface illuminated by a light source, the intensity of illumination on the surface is inversely proportional to the square of the distance between the source and the surface. A certain light source produces an illumination of 800 lux (a lux is 1 lumen per square meter) on a surface. Find the illumination on that surface if the distance to the light source is doubled.

Transformer Turns Ratio: For the transformer shown in Fig. \(17-6,\) the ratio of the number of turns in the secondary winding to the number of turns in the primary winding is \(15 .\) The secondary winding has 4500 turns. Find the number of turns in the primary.

From Eq. \(1066, P=I^{2} R,\) we see that power \(P\) dissipated in a resistor varies directly as the square of the current \(I\) in the resistor. If the current through a resistor is increased by \(28 \%,\) by what percent will the power increase?

The inverse square law states that for a surface illuminated by a light source, the intensity of illumination on the surface is inversely proportional to the square of the distance between the source and the surface. A light source located \(2.75 \mathrm{m}\) from a surface produces an illumination of \(528 \mathrm{lux}\) on that surface. Find the illumination if the distance is changed to \(1.55 \mathrm{m}\).

The amount paid to a work crew varies jointly as the number of persons working and the length of time worked. If 5 workers earn \(5123.73\)dollars in 3.0 weeks, in how many weeks will 6 workers earn a total of \(6148.48\)dollars ?

From Eq. \(1066, P=I^{2} R,\) we see that power \(P\) dissipated in a resistor varies directly as the square of the current \(I\) in the resistor. By what factor must the current in an electric heating coil be increased to triple the power consumed by the heater?

The inverse square law states that for a surface illuminated by a light source, the intensity of illumination on the surface is inversely proportional to the square of the distance between the source and the surface. A light source located \(7.50 \mathrm{m}\) from a surface produces an illumination of 426 lux on that surface. At what distance must that light source be placed to give an illumination of 850 lux?

The exposure time for a photograph is directly proportional to the square of the f stop. (The \(f\) stop of a lens is its focal length divided by its diameter.) A certain photograph will be correctly exposed at a shutter speed of \(1 / 100\) s with a lens opening of \(\mathrm{f} 5.6 .\) What shutter speed is required if the lens opening is changed to \(18 ?\)

The inverse square law states that for a surface illuminated by a light source, the intensity of illumination on the surface is inversely proportional to the square of the distance between the source and the surface. When a document is photographed on a certain copy stand, an exposure time of \(\frac{1}{25} s\) is needed, with the light source 0.750 m from the document. At what distance must the light be located to reduce the exposure time to \(\frac{1}{100} s ?\)

Ideal Mechanical Advantage: Find the ideal mechanical advantage for the screw jack, Fig. \(17-9,\) if the end of the jack handle moves \(57.3 \mathrm{cm}\) to raise the weight \(1.57 \mathrm{cm}.\)

The current in a resistor is inversely proportional to the resistance. By what factor will the current change if a resistance increases \(10.0 \%\) due to heating?

Most cameras have the following \(\mathrm{f}\) stops: \(12.8, \mathrm{f} 4,\) f5.6, \(\mathrm{f} 8, \mathrm{f} 11,\) and \(\mathrm{f} 16\) To keep the same correct exposure, by what factor must the shutter speed be increased when the lens is opened one stop? (Photographer's rule of thumb: Double the exposure time for each increase in \(f\) stop.)

The resistance of a wire is inversely proportional to the square of its diameter. If an AWG (American wire gauge) size 12 conductor \((0.0808\) -in. diameter) has a resistance of \(14.8 \Omega,\) what will be the resistance of an AWG size 10 conductor ( 0.1019 -in. diameter) of the same length and material?

The capacitive reactance \(X_{c}\) of a circuit varies inversely as the capacitance \(C\) of the circuit. If the capacitance of a certain circuit is decreased by \(25.0 \%,\) by what percentage will \(X_{c}\) change?

The kinetic energy of a moving body is directly proportional to its mass and the square of its speed. If the mass of a bullet is halved, by what factor must its speed be increased to have the same kinetic energy as before?

Use a graphics calculator or computer to graph $$y=1 / x \text { and } y=1 / x^{2}$$ in the same viewing window from \(x=-4\) to \(4 .\) How do you explain the different behavior of the two functions for negative values of \(x ?\)

The power available in a jet of liquid is directly proportional to the cross sectional area of the jet and to the cube of the velocity. By what factor will the power increase if the area and the velocity are both increased \(50 \% ?\)