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 \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\)

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

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

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.