A 1.2 -long steel rod with a diameter of 0.50 hangs vertically from the ceiling. An auto engine weighing 4.7 is hung from the rod.
By how much does the rod stretch?
To calculate the amount by which the steel rod stretches under the weight of the auto engine, we need to use Hooke's Law.
Hooke's Law states that the amount a material stretches or deforms is directly proportional to the force applied to it, as long as the material is within its elastic limit. The equation for Hooke's Law is:
F = k * ΔL
Where:
- F is the force applied to the material,
- k is the material's spring constant or stiffness, and
- ΔL is the change or stretch in the length of the material.
In this case, the weight of the auto engine is the force applied, and we need to solve for ΔL.
1. Firstly, let's find the cross-sectional area of the steel rod. The formula for the area of a circle is A = π * r^2, where r is the radius. Since the diameter is given as 0.50 cm, the radius is half of that: 0.50 cm / 2 = 0.25 cm.
2. Convert the radius to meters by dividing by 100: 0.25 cm / 100 = 0.0025 m.
3. Calculate the area: A = π * (0.0025 m)^2.
4. Use the given length to calculate the volume of the steel rod. The formula for volume is V = A * L, where L is the length. V = [π * (0.0025 m)^2] * 1.2 m.
5. To find the weight of the rod itself, we multiply the volume by the density of steel. The density of steel is typically around 7850 kg/m^3.
Weight of rod = [(π * (0.0025 m)^2) * 1.2 m] * 7850 kg/m^3.
6. Calculate the total weight acting on the rod by adding the weight of the engine and the weight of the rod itself: Total weight = Weight of rod + Weight of engine.
7. With the total weight known, we can now solve for ΔL by rearranging the formula as follows: ΔL = F / k.
Here, the force F is the total weight acting on the rod, and k is the spring constant or Young's modulus, a property of the material that represents its stiffness.
The value of the spring constant for steel (Young's modulus) is typically around 200 GPa (Gigapascals) or 200,000,000,000 Pa.
Substituting the values, we can calculate the stretch (ΔL) of the steel rod.