When four people with a combined mass of 320 kg sit down in a car, they find that the car drops .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 (empty) is 2.0 X 10^3 kg?
The spring constant is k = (2000+320)*g/(deflection)
2000 is the mass of the car, and 320 the mass of the occupants.
The frequency of vibration of the empty car is
f = (1/ 2 pi) sqrt (k/2000)
Substitute for k and solve.
2.23 Hz
Sure! Let's break down the problem step by step:
Step 1: Calculate the spring constant (k)
- The spring constant can be calculated using the formula:
k = (2000 + 320) * g / deflection
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and deflection is the drop in height of the car (given as 0.80 cm or 0.0080 m).
- Substitute the values into the formula:
k = (2320) * 9.8 / 0.0080
k = 2858000 N/m (rounded to the nearest whole number)
Step 2: Calculate the frequency of vibration (f) of the empty car
- The frequency can be calculated using the formula:
f = (1 / (2 * pi)) * sqrt(k / 2000)
where pi is a mathematical constant (approximately 3.14159) and k is the spring constant calculated in Step 1.
- Substitute the values into the formula:
f = (1 / (2 * 3.14159)) * sqrt(2858000 / 2000)
f ≈ 7.14 Hz (rounded to two decimal places)
Therefore, the frequency of vibration of the car when it is empty is approximately 7.14 Hz.
To find the frequency of the car's vibration, we need to find the spring constant (k) and the mass of the car (m).
Given:
- Combined mass of the four people = 320 kg
- Deflection of the car when the four people sit in it = 0.80 cm = 0.008 m
- Mass of the car (empty) = 2.0 x 10^3 kg
To find the spring constant (k), we can use the formula:
k = (2000 + 320) * g / deflection
We know that the acceleration due to gravity (g) is approximately 9.8 m/s^2.
Substituting the known values into the formula:
k = (2320) * 9.8 / 0.008
Now, we can calculate the value of k:
k = 284,500 N/m
Next, we can calculate the frequency (f) of the car's vibration using the formula:
f = (1 / 2π) * sqrt(k / m)
Substituting the known values into the formula:
f = (1 / 2π) * sqrt(284500 / 2000)
Now, we can calculate the frequency:
f ≈ 11.25 Hz
Therefore, the frequency of the car's vibration is approximately 11.25 Hz.