Problem 49 - University Physics with Modern Physics

Question

Silica aerogel, an extremely porous, thermally insulating material made of silica, has a density of \(1.00 \mathrm{mg} / \mathrm{cm}^{3}\). A thin circular slice of aerogel has a diameter of \(2.00 \mathrm{~mm}\) and a thickness of \(0.10 \mathrm{~mm}\). a) What is the weight of the aerogel slice (in newtons)? b) What is the intensity and radiation pressure of a \(5.00-\mathrm{mW}\) laser beam of diameter \(2.00 \mathrm{~mm}\) on the sample? c) How many \(5.00-\mathrm{mW}\) lasers with a beam diameter of \(2.00 \mathrm{~mm}\) would be needed to make the slice float in the Earth's gravitational field? Use \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\)

Step-by-Step Solution

Answer

Answer: 10 lasers are needed to make the aerogel slice float in Earth's gravitational field.

Step 1: a) Weight of the aerogel slice

First, we need to find the volume of the aerogel slice. The volume (V) of a cylinder with height (h) and radius (r) is: V = πr^2h The radius is half the diameter, so r = 1 mm. The thickness is the height, so h = 0.1 mm. V = π (1 mm)^2 (0.1 mm) = 0.1π mm^3 Now we'll convert the volume to cubic centimeters, since the given density is in mg/cm³: V = 0.1π (1 cm³ / 1000 mm³) ≈ 3.14 x 10⁻⁴ cm³ Now we can find the mass (m) of the aerogel slice using its density (ρ): m = ρV = (1.00 mg/cm³)(3.14 x 10⁻⁴ cm³) ≈ 3.14 x 10⁻⁴ mg We'll convert this mass to kilograms: m ≈ 3.14 x 10⁻⁴ mg (1 kg / 10⁶ mg) ≈ 3.14 x 10⁻¹⁰ kg Now we can find the weight (W) of the aerogel slice by multiplying its mass by Earth's gravitational acceleration (g): W = mg = (3.14 x 10⁻¹⁰ kg)(9.81 m/s²) ≈ 3.08 x 10⁻⁹ N

Step 2: b) Intensity and radiation pressure of the laser beam

The intensity (I) of a laser beam is the power (P) divided by the area (A) it covers. The area covered by the laser can be found using the formula for the area of a circle (πr²). First, convert the diameter to radius: r = (2.00 mm) / 2 = 1.00 mm = 0.1 cm Now, calculate the area: A = πr² = π (0.1 cm)² = 0.01π cm² Find the intensity: I = P / A = (5 mW) / (0.01π cm²) Remember to convert the power from milliwatts to watts: I = (5 x 10⁻³ W) / (0.01π cm²) ≈ 15.92 W/cm² Now we need to find the radiation pressure (RP) of the laser beam. The formula is: RP = 2I / c where c is the speed of light in vacuum (c ≈ 3.00 × 10^8 m/s). First, convert the intensity (I) to SI units (W/m²): I = 15.92 W/cm² (10^4 cm² / 1 m²) = 1.592 x 10⁵ W/m² Now calculate the radiation pressure: RP = (2)(1.592 x 10⁵ W/m²) / (3.00 × 10^8 m/s) ≈ 1.061 × 10⁻³ N/m²

Step 3: c) Number of lasers to make the slice float

First, let's find the area (A) of the front surface of the aerogel slice exposed to the laser beams. Since the slice is circular and has a diameter of 2.00 mm, we can find the area as: A = πr² = π (1 mm)² = π mm² Convert this area to square meters: A = π mm² (1 m² / 10⁶ mm²) ≈ 3.14 x 10⁻⁶ m² We need to balance the force due to radiation pressure (F_r) with the gravitational force acting on the slice (F_g). The number of lasers (n) needed can be found using the equation: F_r = F_g n(RP)(A) = W n = W / (RP)(A) Substitute the values we found earlier: n ≈ (3.08 x 10⁻⁹ N) / ((1.061 × 10⁻³ N/m²)(3.14 x 10⁻⁶ m²)) ≈ 9.15 Since we cannot have a fraction of a laser, we must round up to the nearest whole number: n = 10 Therefore, 10 lasers are needed to make the slice float in Earth's gravitational field.

Key Concepts

Thermal Insulation

Silica aerogel is renowned for its remarkable thermal insulation properties. Because of its structure, it contains a very high volume of tiny pores. These pores trap air, which has low thermal conductivity, thus impeding heat transfer efficiently. This characteristic makes silica aerogel an excellent insulator.

• The tiny pores in silica aerogel reduce thermal conductivity.
• Because they trap air, they make it difficult for heat to pass through.

Thermal insulation is crucial in applications where maintaining temperature is important. For instance, silica aerogel is often used in spacesuits and building materials where efficient insulation is needed. Its lightweight and flexible nature add to its versatility, making it a preferred choice in various thermal management applications.

Density Calculation

Density is a fundamental concept used to express how much mass is contained within a given volume. In the context of this exercise, the density of silica aerogel is vital to determine its mass and subsequently its weight.

• The formula for density is given by \( \rho = \frac{m}{V} \).
• This formula implies that density is the mass of the object divided by its volume.

To solve for density in practice, you often have to calculate or convert units, as seen in the original exercise. By understanding how to accurately calculate the density, you can further explore other properties like buoyancy and pressure in different environments.

Laser Radiation Pressure

Laser radiation pressure is a fascinating concept in physics where light can exert force on an object. This is important for our problem as it helps understand how radiation could potentially move or levitate objects like the aerogel slice.

• The intensity of the laser is calculated based on the power and area it illuminates.
• Radiation pressure is calculated using the formula \( RP = \frac{2I}{c} \), where \( I \) is intensity and \( c \) is the speed of light.

Understanding this concept is crucial for seeing how light from lasers can apply force. In advanced physics applications, this principle can be used for space propulsion systems or manipulating microscopic particles. It's a favorite technique in discussions about optical tweezers or optomechanical systems.

Gravitational Force Balance

Balancing gravitational forces with other forces, like radiation pressure, is a core idea in many physics problems. When dealing with the gravitational force, it is important to know that it is the product of mass and the gravitational acceleration, given by \( F_g = mg \).

• Earth's gravitational acceleration is approximately \( 9.81 \text{ m/s}^2 \).
• In our problem, we look to counteract \( F_g \) with the radiation force from the lasers.

By equating gravitational force with the force from the laser radiation, we determine how many lasers are necessary to balance and levitate the aerogel slice. This balance of forces is critical in engineering and physics to create stable systems, from aircraft design to space exploration methods.