WHEN SIX PEOPLE WHOSE AVERAGE MASS IS 84 KG SIT DOWN IN A CAR, THEY FIND THAT THE CAR DROPS .84 CM LOWER ON ITS SPRINGS. THEN THEY HAVE GET OUT OF THE CAR AND BOUNCE IT UP AND DOWN. THE ACCELERATION OF GRAVITY IS 9.8 M/S^2. WHAT IS THE FREQUENCY OF THE CAR'S VIBRATION IF ITS MASS (EMPTY) IS 1300 KG? ANSWER IN UNITS OF HZ
To find the frequency of the car's vibration, we can use the relationship between frequency, mass, and displacement in a simple harmonic motion system.
First, let's find the total mass of the car and passengers. We have six people with an average mass of 84 kg, so the total mass of the people is 6 * 84 kg = 504 kg.
The car's mass (empty) is given as 1300 kg.
To calculate the effective mass in the system, we need to include the mass of the car itself and the mass of the people. Therefore, the effective mass is the sum of the empty mass and the mass of the people, which is 1300 kg + 504 kg = 1804 kg.
Next, let's find the gravitational potential energy stored in the car when it is depressed by 0.84 cm. The gravitational potential energy is given by the formula U = mgh, where m is the mass, g is the acceleration due to gravity, and h is the displacement from the equilibrium position.
h = 0.84 cm = 0.84/100 m = 0.0084 m (converting cm to meters)
g = 9.8 m/s^2
Using the formula, U = mgh, the potential energy is U = 1804 kg * 9.8 m/s^2 * 0.0084 m = 146.83968 J.
In simple harmonic motion, the potential energy is equal to the kinetic energy. Therefore, the kinetic energy of the system is also 146.83968 J.
The kinetic energy of a system in simple harmonic motion is given by the equation K = (1/2)mv^2, where m is the mass and v is the velocity.
Let's assume the maximum velocity of the car during vibration is vmax. Therefore, the kinetic energy can be expressed as K = (1/2) * 1804 kg * (vmax)^2.
Since the kinetic energy is equal to the potential energy, we can equate the two expressions:
(1/2) * 1804 kg * (vmax)^2 = 146.83968 J
Now, let's solve for the maximum velocity, vmax:
(vmax)^2 = (2 * 146.83968 J) / 1804 kg
vmax = sqrt((2 * 146.83968 J) / 1804 kg)
Solving this equation, we get vmax ≈ 0.135 m/s.
The frequency (f) of the car's vibration can be calculated using the formula f = (1/2π) * sqrt(k/m), where k is the spring constant and m is the effective mass of the system.
Since we don't have the spring constant in this question, we can rearrange the formula to solve for f:
f = (1/2π) * sqrt(k/m)
f = (1/2π) * sqrt((m * (vmax)^2) / U)
Plugging in the values, we get:
f = (1/2π) * sqrt((1804 kg * (0.135 m/s)^2) / 146.83968 J)
Evaluating this expression, we find:
f ≈ 1.711 Hz
Therefore, the frequency of the car's vibration is approximately 1.711 Hz.