Unit 12: Solids

Small creatures have more surface area per unit of body weight, so they experience greater air resistance per unit of body weight and therefore fall more slowly than humans.

If you double the radius of a sphere, its volume (and weight) increases eight times, while its surface area become only four times larger. Therefore, the surface area per unit body of weight for a small creature, like an insect, is much greater than for a human. As a result, small creatures encounter greater air resistance than humans, resulting in a slower fall and softer impact with the ground.

An open parachute increases the effective surface area of a falling parachutist, which significantly increases the upward force of air resistance. Consequently, once the parachute is open, the parachutist experiences a slower descent toward the ground at a relatively low terminal speed.

12 cm

If a load of 30 N stretches the spring by 6 cm, the force required per unit of distance is expressed as \begin{equation} k = \frac{30\,\text{N}}{6\,\text{cm}} = 5\,\text{N}/\text{cm} \end{equation} where $k$ is the spring constant. In other words, the spring stretches by 1 cm for every 5 N of load. So a 60-N load would stretch the spring by \begin{align} \Delta x &= \frac{F}{k} \nonumber\\[6pt] & = \frac{60\,\text{N}}{5\,\text{N}/\text{cm}} \nonumber\\[6pt] &= 12\,\text{cm} \end{align} as shown in Figure Q12.39.

You disagree.

Liquid water and ice, for example, are both made of H₂O molecules. The difference between the two forms of water is in how the molecules are arranged and bonded. In solids, such as snow crystals, intermolecular attractive forces hold the molecules more tightly and arrange them in a regular pattern.

In glass—an amorphous solid—the silicon atoms are distributed randomly in a noncrystalline (amorphous) structure. In semiconductor devices, the silicon atoms are arranged into a crystalline lattice to form a solid.

It decreases.

When the balloon is heated, the air inside expands, thus increasing the volume of the balloon. Since no air is added or released, the same mass of air now occupies a larger volume. So the density of the air (mass per unit volume) has decreased.

Note that an ice cube does not sink to the bottom of a glass of water.

It decreases. The density of water (at 4°C) is 1000 kg/m³, while the density of ice is 920 kg/m³. This means ice expands when it freezes, which can cause a glass bottle to explode when left in the freezer. This is also why ice cubes float on the surface of a glass of water.

Kindling wood receives more heat per unit mass.

Kindling has more surface area per kilogram than logs and sticks, and most of the mass in small kindling is near the surface. Therefore, it is easier to heat kindling from all sides to the ignition temperature, creating flames that spread quickly. The heat supplied to a log, on the other hand, is not as concentrated, so the log takes longer to reach ignition temperature.

The dome has less surface area per unit volume.

Since heat is lost through the outer walls of a building, the rate at which heat is lost is proportional to the surface area of the building. The external wall of a semispherical (dome-shaped) house has a smaller surface area than the external walls of a conventional block-shaped structure of the same volume.

To minimize heat loss.

Out of all three-dimensional shapes, the ball (or sphere) has the smallest surface area per volume. Therefore, by curling up, animals minimize the rate at which heat leaves their bodies. Note that many animals, like penguins, huddle together to reduce the overall surface area exposed to extremely cold temperatures.

To increase the heat transfer per unit volume.

By making the burger wider and thinner, you increase the surface area for the same volume of meat. As a result, more heat transfers from the barbecue grill to the meat, which causes the burger to cook faster.

The cupcakes will overcook or even burn.

Because of its smaller size, a cupcake has more surface area per volume than a regular cake. The oven heat thus reaches the inside of a cupcake sooner and at a higher rate per unit volume than it would for a regular size cake. Therefore, the baking time for a cake would be too long for the cupcakes.

7.6 g/cm³

The volume of a cylinder of height $h = 10\,\text{cm}$ and radius $r = 5\,\text{cm}$ is calculated as follows: \begin{align} \text{Volume} &= \text{base area} \times \text{height} \nonumber\\[6pt] &= \pi r^2 \times h \nonumber\\[6pt] &= \pi\, (5\,\text{cm})^2 \times 10\,\text{cm} \nonumber\\[6pt] &= 785\,\text{cm}^3 \end{align} The cylinder has a mass of 6 kg (or 6000 g), so you calculate the density of the cylinder as follows: \begin{align} \text{Density} &= \frac{\text{mass}}{\text{volume}} \nonumber\\[6pt] &= \frac{6000\,\text{g}}{785\,\text{cm}^3} \nonumber\\[6pt] &= 7.6\,\text{g}/\text{cm}^3 \end{align}

$2.46\times 10^{10}\,\text{g}$, $10.8\,\text{m}$

The mass of \$700 billion worth of gold is equal to \begin{align} \text{mass} &= \frac{700\times 10^9\,\text{dollar}}{28.40\,\text{dollar/g}} \nonumber\\[6pt] &= 2.46\times 10^{10}\,\text{g} \end{align} This is almost 25,000 metric tons, which is approximately 10 times the world’s gold production in 2009.

The volume of this huge amount of gold (density $= 19.3\,\text{g}/\text{cm}^3$) is calculated as follows: \begin{align} \text{Volume} &= \frac{\text{mass}}{\text{density}} \nonumber\\[6pt] &= \frac{2.46\times 10^{10}\,\text{g}}{19.3\,\text{g}/\text{cm}^3} \nonumber\\[6pt] &= 1.28\times 10^9\,\text{cm}^3 \nonumber\\[6pt] &= 1280\,\text{m}^3 \end{align} If this amount of gold were in the shape of a cube, then you calculate the length of one side by taking the cubic root of the volume: \begin{align} \text{Side length} &= \sqrt[3]{\text{volume}} \nonumber\\[6pt] &= \sqrt[3]{1280\,\text{m}^3} \nonumber\\[6pt] &= 10.8\,\text{m} \end{align}