A 126 kg student sits on a chair which is solely supported by a solid 0.4 meter-long steel rod 0.81 cm in diameter. To the nearest micron (millionth of a meter), what is the change in length of the rod produced by the student's weight?
To find the change in length of the rod produced by the student's weight, we can use Hooke's Law. Hooke's Law states that the change in length of a material is directly proportional to the applied force and inversely proportional to the material's stiffness.
First, we need to calculate the force applied by the student's weight. The weight can be calculated using the formula:
Weight = mass * gravitational acceleration
Given that the student's mass is 126 kg and the gravitational acceleration is approximately 9.8 m/s^2:
Weight = 126 kg * 9.8 m/s^2 = 1234.8 N
Next, we need to calculate the cross-sectional area of the rod. The cross-sectional area of a rod can be calculated using the formula:
Area = pi * (radius)^2
Given that the rod's diameter is 0.81 cm, we need to convert it to meters:
Diameter = 0.81 cm = 0.81 * 10^(-2) m = 0.0081 m
Radius = diameter / 2 = 0.0081 m / 2 = 0.00405 m
Now, we can calculate the cross-sectional area:
Area = pi * (0.00405 m)^2 = 5.17851 * 10^(-5) m^2
Finally, we can calculate the change in length of the rod using the formula:
Change in length = (Force * Length) / (Area * Young's modulus)
Young's modulus is a measure of the stiffness of a material. For steel, Young's modulus is approximately 200 GPa = 200 * 10^9 N/m^2.
Given that the length of the rod is 0.4 m:
Change in length = (1234.8 N * 0.4 m) / (5.17851 * 10^(-5) m^2 * 200 * 10^9 N/m^2)
Change in length = 0.0316604 m ≈ 31.66 microns (millionths of a meter)
Therefore, the change in length of the rod produced by the student's weight is approximately 31.66 microns (to the nearest micron).
To find the change in length of the rod produced by the student's weight, we can use the formula:
ΔL = (F * L) / (π * r^2 * E)
Where:
ΔL is the change in length of the rod
F is the force applied by the student's weight (in Newtons)
L is the original length of the rod (in meters)
r is the radius of the rod (in meters)
E is the Young's modulus of the material (in Pascals)
Let's calculate the values needed for the formula step by step.
1. Convert the student's weight from kilograms to Newtons:
Weight = mass * acceleration due to gravity
Weight = 126 kg * 9.8 m/s^2
2. Convert the length of the rod from centimeters to meters:
Length = 0.4 m
3. Convert the diameter of the rod from centimeters to meters:
Diameter = 0.81 cm = 0.0081 m
Radius (r) = Diameter / 2
4. Look up the Young's modulus of steel. For example, the Young's modulus of steel is approximately 200 GPa (Gigapascals) or 200 × 10^9 Pa.
Now we have all the values we need. Let's substitute them into the formula:
ΔL = (Weight * Length) / (π * r^2 * E)
Calculating the result will give you the change in length of the rod produced by the student's weight. Round the answer to the nearest micron (millionth of a meter).