A spherical steel ball has a diameter of 2.540cm at 25 degree Celsius. (a)What is its diameter when the temperature is raised to 100 degree Celsius? (b)What temperature change is required to increase its volume by 1%?
To answer these questions, we need to know the coefficient of linear expansion for steel. The coefficient of linear expansion (α) for steel is approximately 12 x 10^-6 per degree Celsius.
(a) To find the diameter when the temperature is raised to 100 degrees Celsius, we can use the formula for linear expansion:
ΔL = α * L * ΔT
where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.
Since we are given the diameter and want to find the change in diameter, we need to convert the diameter to the length. The length of a sphere is equal to its diameter, so we can substitute L = D/2 into the formula:
ΔL = α * (D/2) * ΔT
We know the initial diameter is 2.540 cm and we want to find the change in diameter when the temperature is raised from 25 to 100 degrees Celsius. ΔT = 100 - 25 = 75 degrees Celsius.
Substituting the values into the formula, we get:
ΔL = (12 x 10^-6) * (2.540/2) * 75
Simplifying, we have:
ΔL = (12 x 10^-6) * 1.270 * 75
ΔL = 0.01143 cm
Since the change in length is the same as the change in diameter for a sphere, the change in diameter is also 0.01143 cm.
Therefore, the diameter of the steel ball when the temperature is raised to 100 degrees Celsius is approximately 2.540 cm + 0.01143 cm = 2.55143 cm.
(b) To find the temperature change required to increase the volume of the steel ball by 1%, we need to use the coefficient of volume expansion (β) which is three times the coefficient of linear expansion for a solid sphere: β = 3α = 3 * 12 x 10^-6 = 36 x 10^-6 per degree Celsius.
The formula for volume expansion is:
ΔV = β * V * ΔT
where ΔV is the change in volume, β is the coefficient of volume expansion, V is the original volume, and ΔT is the change in temperature.
The volume of a sphere is given by:
V = (4/3) * π * (D/2)^3
We can substitute this into the formula for volume expansion to find the change in volume in terms of the change in diameter:
ΔV = β * (4/3) * π * (D/2)^3 * ΔT
We want to find the temperature change (ΔT) required to increase the volume by 1%, which means ΔV = 0.01 * V.
Substituting these values, we have:
0.01 * V = β * (4/3) * π * (D/2)^3 * ΔT
Since we are given the initial diameter (2.540 cm), we can calculate the initial volume as:
V = (4/3) * π * (2.540/2)^3
Calculating this, we get V ≈ 8.3927 cm^3.
We can now substitute the values into the equation to solve for the temperature change:
0.01 * 8.3927 cm^3 = (36 x 10^-6) * (4/3) * π * (2.540/2)^3 * ΔT
Simplifying further, we have:
0.083927 cm^3 = (36 x 10^-6) * 4 * (π/3) * (1.270/2)^3 * ΔT
0.083927 cm^3 = 4.608 x 10^-8 * ΔT
ΔT = 0.083927 cm^3 / (4.608 x 10^-8)
ΔT ≈ 1823.58 degrees Celsius
Therefore, the temperature change required to increase the volume of the steel ball by 1% is approximately 1823.58 degrees Celsius.
To answer these questions, we need to consider the thermal expansion of the steel ball. When a substance is heated, its particles move faster and the substance expands. The amount of expansion depends on the coefficient of thermal expansion (α) of the material.
(a) First, we need to determine the change in diameter when the temperature is raised from 25 degrees Celsius to 100 degrees Celsius. The formula for linear expansion is given by:
ΔL = Lo * α * ΔT
Where:
ΔL is the change in length,
Lo is the original length,
α is the coefficient of thermal expansion, and
ΔT is the change in temperature.
However, since we are given the diameter, we need to use the formula for the change in diameter:
ΔD = Do * α * ΔT
Where:
ΔD is the change in diameter,
Do is the original diameter (2.540 cm),
α is the coefficient of thermal expansion, and
ΔT is the change in temperature.
To find α for steel, we can look up its coefficient of linear expansion. The coefficient of linear expansion for steel is typically around 10.8 x 10^-6 per °C.
Substituting the values into the formula, we have:
ΔD = 2.540 cm * 10.8 x 10^-6 per °C * (100°C - 25°C)
Simplifying the calculation, we get:
ΔD ≈ 0.00264624 cm
Therefore, the change in diameter when the temperature is raised to 100 degrees Celsius is approximately 0.00264624 cm.
(b) To find the temperature change required to increase the volume by 1%, we need to consider the formula for volume expansion:
ΔV = Vo * β * ΔT
Where:
ΔV is the change in volume,
Vo is the original volume,
β is the coefficient of volume expansion, and
ΔT is the change in temperature.
The coefficient of volume expansion (β) can be calculated from the coefficient of linear expansion (α) using the formula:
β = 3α
So for steel, β ≈ 3 * 10.8 x 10^-6 per °C.
Now, we need to find the change in temperature (ΔT) that would result in a 1% increase in volume. The change in volume can be calculated as a percentage of the original volume:
ΔV = 0.01 * Vo
Substituting the values, we have:
0.01 * Vo = Vo * (3 * 10.8 x 10^-6 per °C) * ΔT
Canceling out Vo, we get:
0.01 = 3 * 10.8 x 10^-6 per °C * ΔT
Simplifying, we find:
ΔT ≈ 0.009259 °C
Therefore, a temperature change of approximately 0.009259°C is required to increase the volume of the steel ball by 1%.