To calculate the compressional stress that the arm withstands during the crash, we need to first calculate the force acting on the forearm bone.
We can use the equation:
F = m * a
where:
F is the force (in Newtons),
m is the effective mass of the arm (in kilograms), and
a is the acceleration (in meters per second squared).
To calculate the acceleration, we can use the equation:
a = (vf - vi) / t
where:
vf is the final velocity (in meters per second),
vi is the initial velocity (in meters per second),
and t is the time taken (in seconds).
Given information:
Initial velocity (vi) = 80 km/h = 80,000 m/3600 s = 22.22 m/s
Final velocity (vf) = 0 m/s (as the arm comes to rest)
Time taken (t) = 5.0 ms = 5.0 * 10^-3 s
Effective mass (m) = 3.0 kg
Using these values, we can calculate the acceleration:
a = (0 - 22.22) / (5.0 * 10^-3)
= -22.22 / 5.0 * 10^-3
= -4444.4 m/s^2 (negative sign indicates deceleration)
Now, we can calculate the force:
F = m * a
= 3.0 kg * -4444.4 m/s^2
= -13333.2 N (negative sign indicates force in the opposite direction)
The compressional stress can be calculated using the formula:
Stress = Force / Area
Given the total cross-sectional area (A) of both forearm bones is approximately 2.0 cm^2, which is equivalent to 2.0 * 10^-4 m^2 (1 cm^2 = 10^-4 m^2 in SI units).
Stress = -13333.2 N / 2.0 * 10^-4 m^2
= -6.67 * 10^7 Pa (approx.)
Therefore, the compressional stress that the arm withstands during the crash is approximately -6.67 * 10^7 Pa. The negative sign indicates a compressive force acting on the forearm bones.