A 2.0-m-tall, 30-cm-diameter concrete column supports a 3.0×105kg load. By how much is the column compressed (in mm) ?
To determine how much the concrete column is compressed, we need to calculate the deformation caused by the load. The column will compress due to the weight of the load acting on it.
First, let's calculate the cross-sectional area of the column. The column is in the shape of a cylinder, so we can use the formula for the area of a circle:
A = πr^2
Given that the diameter of the column is 30 cm, we can calculate the radius (r) by dividing the diameter by 2:
r = 30 cm / 2 = 15 cm
Now, let's convert the radius to meters:
r = 15 cm × 0.01 m/cm = 0.15 m
Using this radius, we can calculate the cross-sectional area of the column:
A = π(0.15 m)^2
Next, we need to calculate the compressive stress (σ) induced by the load on the column. The stress is the force per unit area, given by:
σ = F/A
Here, the force (F) is simply the weight of the load, which is given as 3.0×10^5 kg. Multiplying this by the acceleration due to gravity (9.8 m/s^2), we get the force in Newtons (N).
F = 3.0×10^5 kg × 9.8 m/s^2
Now, we can substitute the values into the formula to calculate the compressive stress (σ).
σ = (3.0×10^5 kg × 9.8 m/s^2) / A
Finally, we can calculate the deformation or compression (ΔL) using Hooke's Law, which states that the deformation of an elastic object is directly proportional to the applied force:
ΔL = (σ × L) / E
In this equation, L is the initial length of the column, which is given as 2.0 m. E is the Young's modulus of concrete, which is typically around 30×10^9 N/m^2.
Substituting the values into the equation, we can calculate the compression (ΔL) in meters. To convert it to millimeters, we can multiply the result by 1000.
Finally, we obtained the answer by performing the necessary calculations.