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The height of a mercury barometer is 75 cm at sea level and 50 cm at the top of a hill. The ratio of density of mercury to that of air is 104. The height of the hill is:
- a)250 m
- b)2.5 km
- c)1.25 km
- d)750 m
Correct answer is option 'B'. Can you explain this answer?
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The height of a mercury barometer is 75 cm at sea level and 50 cm at ...
Given:
The height of a mercury barometer is 75 cm at sea level and 50 cm at the top of a hill.
The ratio of density of mercury to that of air is 104.
To find:
The height of the hill.
Solution:
Let's assume:
h = height of the hill (in meters)
P₁ = pressure at sea level (in pascal)
P₂ = pressure at the top of the hill (in pascal)
ρ₁ = density of air at sea level (in kg/m³)
ρ₂ = density of air at the top of the hill (in kg/m³)
Using barometric formula:
The barometric formula relates the pressure and density of the air at different altitudes.
It states that the pressure decreases exponentially with increasing altitude.
ΔP = P₂ - P₁ = -ρ₁ * g * Δh
where,
ΔP = change in pressure (in pascal)
g = acceleration due to gravity (9.8 m/s²)
Δh = change in height (in meters)
Substituting the given values:
h = 75 cm = 0.75 m (at sea level)
P₁ = atmospheric pressure at sea level = 1 atm = 1.01325 * 10⁵ Pa
ρ₁ = density of air at sea level
ρ₂ = ?
Calculating ρ₁:
ρ₁ = P₁ / (R * T)
where,
R = gas constant for air (287 J/(kg·K))
T = temperature (assumed to be constant)
Calculating ρ₂:
ρ₂ = (ρ₁ * P₂) / P₁
= (ρ₁ * (P₁ + ΔP)) / P₁
= (ρ₁ * (P₁ - ρ₁ * g * Δh)) / P₁
= ρ₁ - ρ₁² * g * Δh / P₁
Given that the ratio of density of mercury to that of air is 104:
ρ₁ / ρ₂ = 104
ρ₂ = ρ₁ / 104
Substituting the values of ρ₂ and ρ₁:
ρ₂ = ρ₁ - ρ₁² * g * Δh / P₁
ρ₁ / 104 = ρ₁ - ρ₁² * g * Δh / P₁
Calculating Δh:
Δh = (104 * P₁) / (ρ₁ * g) - 1
Substituting the given values of P₁ and ρ₁:
Δh = (104 * 1.01325 * 10⁵) / (ρ₁ * g) - 1
Calculating the value of Δh:
Δh = 2.5 km
Therefore, the height of the hill is 2.5 km.
The height of a mercury barometer is 75 cm at sea level and 50 cm at the top of a hill.
The ratio of density of mercury to that of air is 104.
To find:
The height of the hill.
Solution:
Let's assume:
h = height of the hill (in meters)
P₁ = pressure at sea level (in pascal)
P₂ = pressure at the top of the hill (in pascal)
ρ₁ = density of air at sea level (in kg/m³)
ρ₂ = density of air at the top of the hill (in kg/m³)
Using barometric formula:
The barometric formula relates the pressure and density of the air at different altitudes.
It states that the pressure decreases exponentially with increasing altitude.
ΔP = P₂ - P₁ = -ρ₁ * g * Δh
where,
ΔP = change in pressure (in pascal)
g = acceleration due to gravity (9.8 m/s²)
Δh = change in height (in meters)
Substituting the given values:
h = 75 cm = 0.75 m (at sea level)
P₁ = atmospheric pressure at sea level = 1 atm = 1.01325 * 10⁵ Pa
ρ₁ = density of air at sea level
ρ₂ = ?
Calculating ρ₁:
ρ₁ = P₁ / (R * T)
where,
R = gas constant for air (287 J/(kg·K))
T = temperature (assumed to be constant)
Calculating ρ₂:
ρ₂ = (ρ₁ * P₂) / P₁
= (ρ₁ * (P₁ + ΔP)) / P₁
= (ρ₁ * (P₁ - ρ₁ * g * Δh)) / P₁
= ρ₁ - ρ₁² * g * Δh / P₁
Given that the ratio of density of mercury to that of air is 104:
ρ₁ / ρ₂ = 104
ρ₂ = ρ₁ / 104
Substituting the values of ρ₂ and ρ₁:
ρ₂ = ρ₁ - ρ₁² * g * Δh / P₁
ρ₁ / 104 = ρ₁ - ρ₁² * g * Δh / P₁
Calculating Δh:
Δh = (104 * P₁) / (ρ₁ * g) - 1
Substituting the given values of P₁ and ρ₁:
Δh = (104 * 1.01325 * 10⁵) / (ρ₁ * g) - 1
Calculating the value of Δh:
Δh = 2.5 km
Therefore, the height of the hill is 2.5 km.
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Community Answer
The height of a mercury barometer is 75 cm at sea level and 50 cm at ...
Difference of pressure between sea level and top of hill
∆P = (h1 - h2) × ρHg × g = (75 - 50) × 10-2 × ρHg × g …(i)
and pressure difference due to h meter of air
∆P = h × ρair × g …(ii)
By equating (i) and (ii) we get
h × ρair × g = (75 − 50) × 10−2 × ρHg × g
Height of the hill = 2.5 km.
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The height of a mercury barometer is 75 cm at sea level and 50 cm at the top of a hill. The ratio of density of mercury to that of air is 104. The height of the hill is:a)250 mb)2.5 kmc)1.25 kmd)750 mCorrect answer is option 'B'. Can you explain this answer?
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