A car with bad shocks has a mass of 1500 kg. Before you go for a drive with three of your friends you notice that the car sinks a distance of 6.0 cm when all four of you get in the car. You estimate that the four of you together have a mass of 271 kg. As you are driving down the highway at 65 mph you notice that the car is starting to bounce up and down with large amplitude. You realize that there is a periodic series of small bumps and dips in the road that is driving the bouncing. What is the distance (in meters, to two signi�cant �gures) between adjacent bumps on the road, assuming that damping by the shocks is negligible?
Compute the natural frequency of vibration of the shocks with the added mass of passengers. That frequency is
W = [1/(2*pi)]sqrt (k/M) Hz
k is the spring constant, which you can get from the additional deflection due to a known load.
M = 1771 kg
Set the natural frequency equal to the fre
Compute the natural frequency of vibration of the shocks with the added mass of passengers. That frequency is
W = [1/(2*pi)]sqrt (k/M) Hz
k is the spring constant, which you can get from the additional deflection due to a known load.
M = 1771 kg
Set the natural frequency equal to the frequency that bumps are encountered, V/d, and solve for the bump separqation, d.
I did that including making k=(1/2)mvsqared, and I changed 65mph to mps. I get 8.89 as my answer and the computer says I am wrong and should get 36.5 what am I missing
k is the spring constant, NOT the kinetic energy.
k = 271*9.8 N/(0.06 m)= 4.43*10^4 N/m
Thank you so much
To solve this problem, we need to use the concept of simple harmonic motion (SHM).
Let's break down the problem step by step:
1. First, we need to find the effective mass of the car and its passengers. The mass of the car is given as 1500 kg, and the mass of the four passengers is given as 271 kg. So, the total mass is:
Total mass = Mass of car + Mass of passengers
= 1500 kg + 271 kg
= 1771 kg
2. Next, we need to calculate the spring constant of the car's shocks. We can use Hooke's Law, which states that the force exerted by a spring is directly proportional to its spring constant (k) and the displacement (x) from its equilibrium position.
F = k * x
In this case, the displacement (x) is given as 6.0 cm, which is equivalent to 0.06 m. The force required to support the car and passengers can be calculated using the equation:
F = m * g,
where m is the total mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substitute the values to get:
k * x = m * g
k = (m * g) / x
k = (1771 kg * 9.8 m/s^2) / 0.06 m
k = 2874333.3 N/m (to four significant figures)
3. Now that we know the spring constant, we can calculate the angular frequency (ω) of the simple harmonic motion. The angular frequency is given by:
ω = sqrt(k / m),
where k is the spring constant and m is the effective mass (in this case, the total mass).
Substitute the values to get:
ω = sqrt(2874333.3 N/m / 1771 kg)
ω = 14.17 rad/s (to three significant figures)
4. Finally, to find the distance between adjacent bumps on the road, we can use the relationship between the angular frequency (ω) and the period (T) of SHM. The period is the time taken for one complete cycle of motion.
The period is given by:
T = (2π) / ω
Substitute the value of ω to get:
T = (2π) / 14.17 rad/s
T ≈ 0.443 s (to three significant figures)
Since the car is moving at a speed of 65 mph, and the period is the time taken for one complete cycle of motion (in this case, bouncing), we can use the equation:
speed = distance / time
Rearranging the equation, we get:
distance = speed * time
distance = (65 mph * 0.443 s) * (1.609 km/m / 3600 s/hour)
distance ≈ 0.250 km (to three significant figures)
Therefore, the distance between adjacent bumps on the road is approximately 0.250 km (or 250 meters) to two significant figures.