A square plate of sides 10 cm is made of a metal of linear expansivity
2
×
10
-
5
/K. As the plate is heated from 30°C to 100°C, the area of one face of the plate will increase to
The linear expansivity of a material is the fractional increase in its length per degree Celsius. Given a temperature change ΔT, the new length L after heating is related to the initial length L₀ by the formula:
L = L₀ (1 + αΔT)
where α is the linear expansivity.
In this case, the initial side length of the square plate is L₀ = 10 cm, the linear expansivity is α = 2 × 10⁻⁵ /K, and the temperature change is ΔT = 100°C - 30°C = 70°C.
We can plug these values into the formula to find the length of the plate after heating:
L = 10 cm × (1 + 2 × 10⁻⁵ /K × 70 K) = 10 cm × (1 + 0.0014) = 10 cm × 1.0014 = 10.014 cm
The new area A of the square plate will be the square of the new length:
A = L² = (10.014 cm)² ≈ 100.28 cm²
So, the area of one face of the plate will increase to approximately 100.28 cm² after heating.
To calculate the increase in area of the square plate as it is heated from 30°C to 100°C, we need to consider the linear expansivity of the metal and the change in temperature.
The linear expansivity (α) of a material represents how much it expands or contracts for every degree of temperature change. It is usually measured in units of 1/K (inverse Kelvin).
In this case, the linear expansivity of the metal is given as 2 × 10^-5 /K.
First, let's calculate the change in temperature:
Change in temperature (ΔT) = Final temperature - Initial temperature
= 100°C - 30°C
= 70°C
Next, we need to calculate the linear expansion coefficient (ΔL/L), which represents the change in length (ΔL) for every unit length (L) for a given change in temperature.
ΔL/L = α * ΔT
Using the given linear expansivity of 2 × 10^-5 /K and the change in temperature of 70°C, we can calculate the linear expansion coefficient:
ΔL/L = (2 × 10^-5 /K) * 70°C
Now, we know that the area of a square is given by the formula A = side^2, where "side" represents the length of one side of the square.
Since we have a square plate with sides measuring 10 cm, the initial area (A_initial) is:
A_initial = (10 cm)^2
= 100 cm^2
To calculate the increase in area (ΔA), we need to multiply the initial area by the linear expansion coefficient:
ΔA = A_initial * ΔL/L
Substituting the values:
ΔA = 100 cm^2 * (2 × 10^-5 /K) * 70°C
Now, we can calculate the final area (A_final) by adding the increase in area to the initial area:
A_final = A_initial + ΔA
Substituting the values:
A_final = 100 cm^2 + 100 cm^2 * (2 × 10^-5 /K) * 70°C
Now, let's calculate the final area:
A_final = 100 cm^2 + 100 cm^2 * (2 × 10^-5 /K) * 70°C
A_final = 100 cm^2 + 0.014 cm^2
A_final = 100.014 cm^2
Therefore, the area of one face of the square plate will increase to approximately 100.014 cm^2.
To find the increase in area of one face of the plate, we can use the formula:
ΔA = 2A₀αΔT
Where:
ΔA is the change in area
A₀ is the initial area of the face of the plate
α is the linear expansivity of the metal
ΔT is the change in temperature
Given:
Side of the square plate, s = 10 cm
Initial temperature, T₀ = 30°C
Final temperature, T = 100°C
Linear expansivity, α = 2 × 10^(-5) /K
First, let's calculate the initial area, A₀, of one face of the plate:
A₀ = s² = (10 cm)^2 = 100 cm²
Next, let's calculate the change in temperature, ΔT:
ΔT = T - T₀ = 100°C - 30°C = 70°C
Now, we can plug in the values into the formula to find the change in area, ΔA:
ΔA = 2A₀αΔT
ΔA = 2(100 cm²)(2 × 10^(-5) /K)(70°C)
ΔA = 0.0004 cm²/K * 70°C
ΔA = 0.028 cm²
Therefore, as the plate is heated from 30°C to 100°C, the area of one face of the plate will increase by 0.028 cm².