Four people, each with a mass of 72.7 kg, are in a car with a mass of 1110 kg. An earthquake strikes. The driver manages to pull of the road and stop, as the vertical oscillations of the ground surface make the car bounce up and down on its suspension springs. When the frequency of the shaking is 1.80 Hz, the car exhibits a maximum amplitude of vibration. The earthquake ends and the four people leave the car as fast as they can. By what distance does the car's undamaged suspension lift the car's body as the people get out?
To find the distance by which the car's undamaged suspension lifts the car's body as the people get out, we need to calculate the maximum displacement of the car during the earthquake.
Given:
Mass of each person (m) = 72.7 kg
Mass of the car (M) = 1110 kg
Frequency of shaking (f) = 1.80 Hz
First, let's calculate the angular frequency (ω) of the oscillation using the formula:
ω = 2πf
Plugging in the given frequency, we have:
ω = 2π * 1.80 Hz = 3.60π rad/s
Next, we can calculate the angular velocity (v) of the oscillation using the formula:
v = ω * amplitude
Since the maximum amplitude is not given, we will need to use another formula to find it. The maximum displacement (x) of the oscillation can be found using the formula:
x = F / (k * m)
where F is the force applied on the suspension, k is the spring constant, and m is the total mass (mass of the car plus the people).
The force applied on the suspension can be calculated using Newton's second law of motion:
F = m * a
where m is the total mass and a is the acceleration.
Now, to find the acceleration, we can use the formula for the net force on the car:
F = M * a
where M is the mass of the car.
Adding the forces due to the people and the car, we have:
F = (4 * m + M) * a
Substituting the given values, we get:
F = (4 * 72.7 kg + 1110 kg) * a
Finally, substituting this expression for F back into the formula for x, we find:
x = [(4 * 72.7 kg + 1110 kg) * a] / (k * m)
Given that the car exhibits a maximum amplitude of vibration during the earthquake, the displacement x is the amplitude of the oscillation.
Note: To calculate the distance by which the car's undamaged suspension lifts the car's body as the people get out, we would need to determine the spring constant (k) and the acceleration (a) during the earthquake, which are not provided in the question.