Answer:
A λ = 97.23 nm
, B) λ = 486.2 nm
, C) λ = 53326 nm
Explanation:
With that problem let's use the Bohr model equation for the hydrogen atom
= -k e² /2a₀ 1/n²
For a transition between two states we have
-
= -k e² /2a₀ (1/
² - 1 / n₀²)
Now this energy is given by the Planck equation
E = h f
And the speed of light is
c = λ f
Let's replace
h c / λ = - k e² /2a₀ (1 /
² - 1 / no₀²)
1 / λ = - k e² /2a₀ hc (1 /
² -1 / n₀²)
Where the constants are the Rydberg constant
= 1.097 10⁷ m⁻¹
1 / λ =
(1 / n₀² - 1 / nf²)
Now we can substitute the given values
Part A
Initial state n₀ = 1 to the final state
= 4
1 / λ = 1.097 10⁷ (1/1 - 1/4²)
1 / λ = 1.0284 10⁷ m⁻¹
λ = 9.723 10⁻⁸ m
We reduce to nm
λ = 9.723 10⁻⁸ m (10⁹ nm / 1m)
λ = 97.23 nm
Part B
Initial state n₀ = 2 final state
= 4
1 / λ = 1.097 10⁷ (1/2² - 1/4²)
1 / λ = 0.2056 10⁻⁷ m
λ = 486.2 nm
Part C
Initial state n₀ = 3
1 / λ = 1,097 10⁷ (1/3² - 1/4²)
1 / λ = 5.3326 10⁵ m⁻¹
λ = 5.3326 10-5 m
λ = 53326 nm
The concepts used to solve this exercise are given through the calculation of distances (from the Moon to the earth and vice versa) as well as the gravitational potential energy.
By definition the gravitational potential energy is given by,

Where,
m = Mass of Moon
G = Gravitational Universal Constant
M = Mass of Ocean
r = Radius
First we calculate the mass through the ratio given by density.



PART A) Gravitational potential energy of the Moon–Pacific Ocean system when the Pacific is facing away from the Moon
Now we define the radius at the most distant point

Then the potential energy at this point would be,



PART B) when Earth has rotated so that the Pacific Ocean faces toward the Moon.
At the nearest point we perform the same as the previous process, we calculate the radius

The we calculate the Potential gravitational energy,



Answer
given,
Weight of the child = 110 N
length of the swing,L = 2 m
now, calculating the potential energy when the string is horizontal
Potential energy = m g h
now, h = L (1 - cos θ) where θ is the angle made by the string with the vertical.
PE = m g L (1 - cos θ)
when rope is horizontal θ = 90°
PE = 110 x 2 (1 - cos 90°)
PE = 220 J
now, calculating potential energy when string made 25° with horizontal
PE = m g L (1 - cos θ)
when rope is horizontal θ = 25°
PE = 110 x 2 (1 - cos 25°)
PE = 20.61 J
The final velocity of the object is 16m\s.
Hope this helps! :)