A car with a total mass of 1800kg (including passengers) is driving down a washboard road with bumps spaced 5.0m apart. The ride is roughest—that is, the car bounces up and down with the maximum amplitude—when the car is traveling at 6.1 m/s.What is the spring constant of the car’s springs?
what is the period when the resonance occurs?
T = distance/speed
= 5 / 6.1 m
seconds
so frequency f = 1/T = 6.1/5
Now do your sqrt K/m thing
To find the spring constant of the car's springs, we can use the concept of simple harmonic motion. In this case, we can treat the car as a mass-spring system, where the bumps on the road act as the restoring force.
First, we need to calculate the angular frequency (ω) of the car's motion. The angular frequency can be determined using the following formula:
ω = 2πf,
where f is the frequency of the motion. The frequency can be calculated as the reciprocal of the period (T), which is the time taken for one complete oscillation:
T = 1/f.
In this case, we know that the bumps are spaced 5.0 m apart. Therefore, the distance traveled by the car in one complete oscillation is twice the distance between the bumps:
Distance per oscillation = 2 * 5.0 m = 10.0 m.
The time taken for one complete oscillation, the period, can then be found by dividing the distance by the car's velocity:
T = Distance per oscillation / Velocity = 10.0 m / 6.1 m/s.
Now that we have the period (T), we can calculate the frequency (f):
f = 1 / T.
Finally, we can calculate the angular frequency (ω):
ω = 2πf.
Once we have the angular frequency (ω), we can use it to find the spring constant (k) of the car's springs.
The formula to find the spring constant is given by:
k = mω^2,
where m is the mass of the car (1800 kg).
By substituting the values, we can calculate the spring constant (k) of the car's springs.
To calculate the spring constant of the car's springs, we can use the formula for the natural frequency of a mass-spring system. The natural frequency can be calculated using the formula:
f = (1/2π) * sqrt(k/m)
where:
f is the natural frequency,
k is the spring constant, and
m is the mass.
Given:
Total mass of the car (including passengers), m = 1800 kg
Speed of the car, v = 6.1 m/s
Distance between bumps, d = 5.0 m
First, we need to calculate the time period of the bumps. The time period can be calculated using the formula:
T = d/v
where:
T is the time period.
T = 5.0 m / 6.1 m/s
T ≈ 0.82 s
Next, we can calculate the frequency of the bumps:
f = 1/T
f = 1 / 0.82 s
f ≈ 1.22 Hz
Now, we can use the natural frequency formula to find the spring constant:
f = (1/2π) * sqrt(k/m)
1.22 Hz = (1/2π) * sqrt(k / 1800 kg)
Rearranging the formula, we can solve for k:
k = (4π² * m * = (4 * 3.14² * 1800 kg * (1.22 Hz)²)
k ≈ 46204.95 N/m
Therefore, the spring constant of the car's springs is approximately 46204.95 N/m.