An automobile can be considered to be mounted on four identical springs as far as vertical oscillations are concerned. The springs of a certain car are adjusted so that the oscillations have a frequency of 4 Hz.
(a) What is the spring constant of each spring if the mass of the car is 1450 kg and the weight is evenly distributed over the springs?
[[[[ i figured this out.. and the answer is: 2.28e5 N/m ]]]]
(b) What will be the vibration frequency if five passengers, averaging 77.0 kg each, ride in the car with an even distribution of mass? ___ Hz
.. im not sure what im doing wrong here..
sqrt((2.28e5)/(1450+5(77))) = 11.14 rads/s
11.14 / 2pi = 1.75 Hz
.. should i be adding the weight of the passengers including the weight of the car on the springs? .. i feel like im soo close to the answer, but i cant grasp what's wrong. please help!
sqrt (k/m) = w
k= w^2 * m= (16*4*PI^2)*1450
k of each spring is k/4=16PI^2*1450= 2.29E5 N/m
Now change the mass
w= sqrt k/m= sqrt (2.29E5*4/1835)=41.35
f= w/2PI=6.58 hz
check that.
To calculate the vibration frequency when five passengers, averaging 77.0 kg each, ride in the car, you need to consider the additional mass added to the system. Here's how you can calculate it:
1. Find the total mass of the system: The mass of the car is already given as 1450 kg. Since there are five passengers, each weighing an average of 77.0 kg, the total mass of the passengers is 5 * 77.0 kg = 385 kg.
2. Calculate the combined mass: Add the total mass of the car and the passengers to find the combined mass of the system: 1450 kg + 385 kg = 1835 kg.
3. Calculate the spring constant: You already found the spring constant of each spring in part (a) to be 2.28e5 N/m.
4. Calculate the angular frequency: Use the formula w = sqrt(k/m), where k is the spring constant and m is the combined mass.
w = sqrt((2.28e5 N/m) / (1835 kg)) = 11.136 rad/s
5. Calculate the vibration frequency: Divide the angular frequency by 2π to get the vibration frequency in Hz.
f = w / (2π) = 11.136 rad/s / (2π) = 1.768 Hz (approximately)
So the vibration frequency when five passengers are riding in the car with an even distribution of mass is approximately 1.768 Hz.