We hang a mass of 500kg on a cylindrical steel rod of radius 0.5 cm and length 1 m. How much will the rod stretch? Young's modulus: steel 12*10^10 N/m^2.
Now change steel to nylon. How much will it stretch?
Nylon: 0.36*10^10 N/m^2.
Thanks.
W/A =Y(ΔL/L)
ΔL=W•L/A•Y=
=mg•L/πr²•Y
To determine how much the rod will stretch, we can use Hooke's Law, which states that the extension or stretch of an object is directly proportional to the applied force.
First, let's calculate the cross-sectional area of the cylinder using the given radius (0.5 cm) and the formula for the area of a circle:
A = π * r^2
Converting the radius to meters: 0.5 cm = 0.005 m
A = π * (0.005)^2 = 0.00007854 m^2
Next, we can calculate the force applied to the rod by multiplying the mass (500 kg) by the acceleration due to gravity (9.8 m/s^2):
F = m * g = 500 kg * 9.8 m/s^2 = 4900 N
Now, we can use Hooke's Law to calculate the stretch or extension of the rod. Hooke's Law equation is given by:
ΔL = (F * L) / (A * Y)
where:
ΔL is the change in length or stretch
F is the force applied
L is the original length of the rod
A is the cross-sectional area of the rod
Y is Young's modulus for the material
For steel:
ΔL_steel = (4900 N * 1 m) / (0.00007854 m^2 * 12 * 10^10 N/m^2)
Calculating this gives us:
ΔL_steel ≈ 0.524 mm
Therefore, the steel rod will stretch by approximately 0.524 mm when a mass of 500 kg is hung on it.
Now, let's calculate the stretch when the material is changed to nylon, which has a Young's modulus of 0.36 * 10^10 N/m^2. Using the same formula:
ΔL_nylon = (4900 N * 1 m) / (0.00007854 m^2 * 0.36 * 10^10 N/m^2)
Calculating this gives us:
ΔL_nylon ≈ 16.8 mm
Therefore, when the material is changed to nylon, the rod will stretch by approximately 16.8 mm.
Please note that these calculations assume the rod is perfectly elastic and obeys Hooke's Law, which may not be completely accurate in real-life situations.