When four people with a combined mass of 320 kg sit down in a car, they find that the car drops 0.80 cm lower on its springs. Then they get out of the car and bounce it up and down. What is the frequency of the car's vibration if its mass (when it is empty) is 2.0 x 10^3 kg?
2.25
7.05 Hz
To find the frequency of the car's vibration, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
First, let's calculate the change in potential energy when the four people sit in the car and cause it to drop 0.80 cm. The potential energy stored in a spring is given by the formula:
U = (1/2)kx^2
Where U is the potential energy, k is the spring constant, and x is the displacement.
Since the car drops 0.80 cm lower on its springs, we can convert this to meters by dividing by 100:
x = 0.80 cm ÷ 100 = 0.008 m
Now, let's calculate the spring constant using the change in potential energy and the displacement:
U = (1/2)kx^2
Solving for k:
k = (2U) / x^2
Since U is the change in potential energy, we can assume it is equal to the work done against gravity:
U = mgh
Where m is the mass, g is the acceleration due to gravity, and h is the height.
Given that there is a combined mass of 320 kg, we can calculate m:
m = 320 kg
The acceleration due to gravity is approximately 9.8 m/s^2, and the change in height is 0.008 m:
U = (320 kg)(9.8 m/s^2)(0.008 m)
Now we can calculate the spring constant:
k = (2U) / x^2
k = (2(320 kg)(9.8 m/s^2)(0.008 m)) / (0.008 m)^2
Once we have the spring constant, we can use it to calculate the frequency of the car's vibration when it is empty. The frequency of a vibrating system is given by the formula:
f = (1 / (2π)) * √(k/m)
Where f is the frequency, k is the spring constant, and m is the mass.
Given that the mass of the car when it is empty is 2.0 x 10^3 kg, we can calculate the frequency:
f = (1 / (2π)) * √(k/m)
f = (1 / (2π)) * √(k / (2.0 x 10^3 kg))
By plugging in the values of k and m, we can calculate the frequency of the car's vibration.
Use the additional compression distance (0.0080 m), due to added tha weight of 320 kg*g, to determine the spring constant, k, in units of N/m.
k = 3136 N/.0008 m = ____
The frequency of car vibration is then
f = [1/(2 pi)]*sqrt(k/M)
where M = 2000 kg