The total cross-sectional area of the load-bearing calcified portion of the two forearm bones (radius and ulna) is approximately 2.0 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 5.0 ms. If the arm has an effective mass of 3.0 kg, what is the compressional stress that the arm withstands during the crash?
Pa
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.
To calculate the compressional stress that the arm withstands during the crash, we can use the formula for stress:
Stress = Force / Area
First, let's calculate the force exerted on the forearm during the crash. We can use the equation:
Force = Mass x Acceleration
We know the mass of the arm (m = 3.0 kg) and the time it takes to come to rest (t = 5.0 ms = 0.005 s). We can calculate the acceleration using the equation:
Acceleration = Change in Velocity / Time
Given the initial velocity (v = 80 km/h) and the time (t = 0.005 s), we can convert the velocity to m/s and calculate the change in velocity:
Initial velocity (v) = 80 km/h = (80 * 1000) / 3600 = 22.22 m/s
Change in velocity (Δv) = 0 m/s (since the arm comes to rest)
Now we can calculate the acceleration:
Acceleration = (Change in velocity) / (Time) = (0 m/s - 22.22 m/s) / 0.005 s = -4444 m/s²
Now we can calculate the force exerted on the forearm:
Force = (Mass) x (Acceleration) = (3.0 kg) x (-4444 m/s²) = -13,332 N
Note that the force is negative because it acts in the opposite direction of the initial motion.
Finally, we can calculate the compressional stress:
Stress = (Force) / (Area) = (-13,332 N) / (2.0 cm²) = -6,666 Pa
The compressional stress that the arm withstands during the crash is approximately -6,666 Pa.