The body of a 1290 kg car is supported on a
frame by four springs. The spring constant of
a single spring is 2.59 × 104 N/m. Four peo-
ple riding in the car have a combined mass
of 246 kg. When driven over a pothole in
the road, the frame vibrates and for the first
few seconds the vibration approximates sim-
ple harmonic motion.
What is the period of vibration of the car?
Answer in units of s
To find the period of vibration of the car, we can use the formula:
Period (T) = 2π * √(m/k)
Where:
- m is the effective mass supported by the springs, which is the sum of the car's mass (1290 kg) and the combined mass of the people riding in the car (246 kg).
- k is the spring constant of a single spring (2.59 × 10^4 N/m).
Let's plug in the values and calculate the period:
m = 1290 kg + 246 kg = 1536 kg
k = 2.59 × 10^4 N/m
T = 2π * √(1536 kg / (2.59 × 10^4 N/m))
≈ 2π * √(0.0592)
Now, we can calculate the approximate value of the period:
T ≈ 2π * 0.243
≈ 1.527 seconds
Therefore, the period of vibration of the car is approximately 1.527 seconds.
To find the period of vibration of the car, we need to know the effective spring constant of the system and the total mass of the car and passengers.
The effective spring constant of the system is given by the equation:
k_eff = (k1 + k2 + k3 + k4)
where k1, k2, k3, and k4 are the individual spring constants. In this case, all four springs have the same spring constant of 2.59 × 104 N/m, so we can simplify the equation:
k_eff = 4 * (2.59 × 104 N/m) = 1.036 × 105 N/m
Next, we need to find the total mass of the car and passengers:
m_total = m_car + m_passengers
where m_car is the mass of the car and m_passengers is the combined mass of the four people. We are given that the car has a mass of 1290 kg and the passengers have a combined mass of 246 kg, so:
m_total = 1290 kg + 246 kg = 1536 kg
Now we can calculate the period of vibration using the formula:
T = 2π * √(m_total / k_eff)
where T is the period of vibration.
Plugging in the values we found:
T = 2π * √(1536 kg / 1.036 × 105 N/m)
Calculating the value, we get:
T ≈ 2π * √(0.01482)
Finally, we simplify the equation to find the period of vibration:
T ≈ 6.907 s
Therefore, the period of vibration of the car is approximately 6.907 seconds.
(1290+246)/4
Then use the formula T=2*pi**squareroot*(m/k) to find the answer