Question

A totally reflecting disk has radius 8. 00 um, thickness 2. 00 um, and average density 7. 00x102 kg/m². A laser has an average power output Pav spread uniformly over a cylindrical beam of radius 2. 00 mm. When the laser beam shines upward on the disk in a direction perpendicular to its flat surface, the radiation pressure produces a force equal to the weight of the disk. Part A What value of Pay is required? Express your answer with the appropriate units. ΜΑ ?

Answer 1

2.69 × 10^(-10) N

Step-by-step explanation:

To calculate the required value of power, we need to consider the force exerted by the radiation pressure and equate it to the weight of the disk.

The force exerted by the radiation pressure is given by:

F = (2RΔt)P/c

where:

F is the force,

R is the radius of the disk,

Δt is the thickness of the disk,

P is the power of the laser,

c is the speed of light.

We are given:

R = 8.00 μm = 8.00 × 10^(-6) m (radius of the disk)

Δt = 2.00 μm = 2.00 × 10^(-6) m (thickness of the disk)

ρ = 7.00 × 10^2 kg/m² (average density of the disk)

The weight of the disk is given by:

W = mg

where:

m is the mass of the disk,

g is the acceleration due to gravity.

The mass of the disk can be calculated using its average density and volume:

m = ρV

The volume of the disk is given by:

V = πR²Δt

Substituting the expressions for mass and volume into the equation for weight, we have:

W = ρVg = ρ(πR²Δt)g

Setting the force equal to the weight, we have:

F = W

(2RΔt)P/c = ρ(πR²Δt)g

Simplifying the equation:

2RP/c = ρπR²g

Now we can solve for the power P:

P = (ρπRg)/(2c)

Substituting the given values:

P = (7.00 × 10^2 kg/m²)(π)(8.00 × 10^(-6) m)(9.8 m/s²)/(2(3.00 × 10^8 m/s))

Calculating this expression:

P ≈ 2.69 × 10^(-10) kg⋅m/s² = 2.69 × 10^(-10) N

So, the required power Pay is approximately 2.69 × 10^(-10) N (newtons).