The total cross-sectional area of the load-bearing calcified portion of the two forearm bones (radius and ulna) is approximately 2.5 cm2. During a car crash, the forearm is slammed against the dashboard. The arm comes to rest from an initial speed of 80 km/h in 4.2 ms. If the arm has an effective mass of 3.0 kg, what is the compressional stress that the arm withstands during the crash?
To calculate the compressional stress that the arm withstands during the crash, we can use the formula:
Stress = Force / Area
First, let's calculate the force exerted on the arm during the crash. We can use Newton's second law, which states that force is equal to mass multiplied by acceleration:
Force = mass * acceleration
The acceleration can be calculated using the formula for average acceleration:
Acceleration = (final velocity - initial velocity) / time
Convert the initial velocity from km/h to m/s:
Initial velocity = 80 km/h * (1000 m/1 km) * (1 h/3600 s) = 22.22 m/s
Substituting the values into the formula:
Acceleration = (0 - 22.22 m/s) / 4.2 ms = -5276.19 m/s^2
Now, substitute the values for mass and acceleration into the formula for force:
Force = 3.0 kg * -5276.19 m/s^2 = -15828.57 N
Since the compressional stress refers to force per unit area, we can substitute force and area into the formula for stress:
Stress = -15828.57 N / 2.5 cm^2 = -6331.43 Pa
The compressional stress that the arm withstands during the crash is approximately -6331.43 Pa. Note that stress is a scalar quantity, so the negative sign indicates a compressional stress.
To calculate the compressional stress that the arm withstands during the crash, we need to know the force exerted on the forearm and the cross-sectional area. Since we are given the total cross-sectional area of the two forearm bones (radius and ulna) as 2.5 cm², we can begin by converting it to square meters:
1 cm² = 0.0001 m²
Therefore, the cross-sectional area in square meters is:
2.5 cm² * 0.0001 m²/cm² = 0.00025 m²
Next, we need to calculate the force exerted on the forearm during the crash. To do this, we can use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a):
F = m * a
We are given that the arm has an effective mass of 3.0 kg and comes to rest from an initial speed of 80 km/h in 4.2 ms. To calculate the acceleration, we need to convert the initial speed from km/h to m/s and divide it by the time it takes for the arm to come to rest:
80 km/h = 80 * 1000 m/3600 s = 22.22 m/s
4.2 ms = 4.2 * 10⁻³ s
Now we can calculate the acceleration:
a = (final velocity - initial velocity) / time
= (0 - 22.22 m/s) / 4.2 * 10⁻³ s
= -22.22 m/s / 4.2 * 10⁻³ s
= -5280.95 m/s²
Since the arm comes to rest, the acceleration is negative. However, compressional stress is a scalar value and does not include the sign. Therefore, we can ignore the negative sign and use the magnitude of the acceleration:
a = 5280.95 m/s²
Now we can calculate the force exerted on the forearm:
F = m * a
= 3.0 kg * 5280.95 m/s²
= 15842.85 N
Finally, we can calculate the compressional stress by dividing the force by the cross-sectional area of the forearm:
Compressional Stress = Force / Area
= 15842.85 N / 0.00025 m²
= 63,371,400 N/m²
Therefore, the compressional stress that the arm withstands during the crash is approximately 63,371,400 N/m².