The body of a 1297 kg car is supported on a frame by four springs. The spring constant of

a single spring is 1.24 × 10
4
N/m. Four people riding in the car have a combined mass
of 259 kg. When driven over a pothole in
the road, the frame vibrates and for the first
few seconds the vibration approximates simple harmonic motion.
What is the period of vibration of the car?
Answer in units of s

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To find the period of vibration of the car, we can use the formula for the period of a mass-spring system:

T = 2π√(m/k)

Where:
T = Period of vibration
π = Pi (approximately 3.14159)
m = Effective mass (mass of the car plus the combined mass of the people)
k = Spring constant (given as 1.24 × 10^4 N/m)

Let's calculate the effective mass of the system first:

Effective mass = mass of the car + combined mass of the people
Effective mass = 1297 kg + 259 kg
Effective mass = 1556 kg

Now we can substitute the values into the period equation:

T = 2π√(m/k)
T = 2π√(1556 kg / 1.24 × 10^4 N/m)

Let's calculate the period:

To find the period of vibration of the car, we can use the formula:

T = 2π√(m/k)

where T is the period, m is the mass, and k is the spring constant. In this case, the total mass of the car and the people riding in it is given as 1297 kg.

However, we need to modify the mass value to account for the effective mass supported by the springs. When the car goes over a pothole, the springs compress and store potential energy, which manifests as vibrations. This effectively increases the mass supported by the springs.

The modified mass can be calculated by adding the combined mass of the people to the mass of the car and dividing by the number of springs. In this case, there are four springs.

Modified mass = (Mass of car + Mass of people) / Number of springs
= (1297 kg + 259 kg) / 4
= 1556 kg / 4
= 389 kg

Now we can substitute the values into the formula:

T = 2π√(m/k)
= 2π√(389 kg / (1.24 × 10^4 N/m))

To calculate the period, we can use a calculator or a computer to evaluate this expression:

T ≈ 2π√(389 / 1.24 × 10^4)
≈ 2π√(389 / 12400)
≈ 2π√0.03137
≈ 2π × 0.177

Now we can calculate the numerical value:

T ≈ 2π × 0.177
≈ 1.114 s

Therefore, the period of vibration of the car is approximately 1.114 seconds.

The combined spring constant of the four springs "in parallel" is four times that of a sintle spring, which is

k = 4.96*10^4 N/m

m is the total mass of car and occupants, which appears to be 1556 km.

The period of vibration is 2*pi*sqrt(m/k)
Do the calculation.