The mass of an unloaded car is 800 kg. The body of the car will sink 6 cm after getting in 5 persons of total mass of 500 kg. How much is the time of period of vibration of the unloaded car and loaded with passengers?
To find the time period of vibration for the unloaded car and the loaded car with passengers, we need to consider the concept of a spring-mass system.
1. Calculate the effective mass of the unloaded car when it sinks.
The effective mass of the unloaded car is given by the formula:
m_unloaded = m_car + m_spring
Where:
m_car = mass of the car
m_spring = mass equivalent to the body sinking due to the car's suspension
Given:
Mass of the car (m_car) = 800 kg
Body sinking (d) = 6 cm = 0.06 m
We can calculate the equivalent mass using the formula:
m_spring = k * d / g
Where:
k = spring constant
g = acceleration due to gravity (approximately 9.8 m/s^2)
However, we don't have information about the spring constant for the car's suspension, so we cannot calculate the effective mass accurately. We'll assume for simplicity that the spring constant is 1 N/m, which is a typical value for car suspensions.
m_spring = 1 * 0.06 / 9.8
m_spring ≈ 0.0061 kg
m_unloaded = m_car + m_spring
m_unloaded ≈ 800 + 0.0061
m_unloaded ≈ 800.0061 kg
2. Calculate the effective mass of the loaded car with passengers.
m_loaded = m_unloaded + m_passengers
Given:
Total mass of the passengers (m_passengers) = 500 kg
m_loaded = 800.0061 + 500
m_loaded ≈ 1300.0061 kg
3. Calculate the time period of vibration for the loaded and unloaded car.
The time period of vibration for a spring-mass system is given by the formula:
T = 2π * √(m / k)
Where:
T = time period of vibration
m = mass of the system
k = spring constant
For the unloaded car:
T_unloaded = 2π * √(m_unloaded / k)
For the loaded car with passengers:
T_loaded = 2π * √(m_loaded / k)
As we assumed the spring constant to be 1 N/m, we can combine the formulas to calculate the two time periods:
T_unloaded ≈ 2π * √(800.0061 / 1)
T_loaded ≈ 2π * √(1300.0061 / 1)
Calculating these values will give you the time period of vibration for the unloaded car and the loaded car with passengers.