in a certain automobile when five passenger get in their total mass being 360kg the spring of the suspension are compressed a distance 4cm. the total mass including the passenger supported by the suspension is 900kg. assuming that the displacement of the spring is directly proportional to the compressive force in them calculate the period of oscillation
weight of passengers = 360 * 9.81 = 3532 Newtons
distance they compressed spring = x = 0.04 meters
F = kx
3532 = k (0.04)
k = 88,290 Newtons/meter
then assume x = A sin (2 pi t/T)
then
v = A(2 pi/T) cos (2 pi t/T)
a = -A (2 pi / T)^2 sin (2 pi t/T) = - (2 pi /T)^2 x
then
-F = m a (- F because force by spring on mass, not by weight on spring)
- k x = - 900(2 pi /T)^2
-88,290 = - 900(2 pi/T)^2
2 pi/T =9.90
T = 0.634 second
To calculate the period of oscillation, we first need to find the spring constant (k). Here's how you can do it:
1. Start by finding the initial compression of the springs when the five passengers get in the automobile. The total mass of the passengers is 360 kg, and the displacement of the springs is 4 cm (0.04 m).
2. Use Hooke's Law to calculate the force applied by the passengers on the springs. According to Hooke's Law, the force applied to a spring is directly proportional to its displacement. The formula for Hooke's Law is F = -kx, where F is the force, k is the spring constant, and x is the displacement.
3. Rearrange the formula to solve for the spring constant: k = -F / x. Since we know the force applied (-F) is equal to the weight of the passengers, we can calculate it using the gravitational acceleration (9.8 m/s^2) and the total mass (360 kg).
- F = mg, where m is the total mass of the passengers.
4. Calculate the spring constant using the formula k = -F / x.
Once we have the spring constant, we can calculate the period of oscillation using the formula:
T = 2π√(m / k),
where T is the period, m is the total mass including the passengers (900 kg), and k is the spring constant.
Let's calculate the spring constant and then find the period of oscillation:
Step 1: Calculate the force applied by the passengers on the springs.
F = mg = (360 kg) * (9.8 m/s^2) = 3528 N
Step 2: Calculate the spring constant.
k = -F / x = -3528 N / 0.04 m = -88200 N/m
Step 3: Calculate the period of oscillation.
T = 2π√(m / k) = 2π√(900 kg / -88200 N/m) ≈ 0.654 s
Therefore, the period of oscillation is approximately 0.654 seconds.