The shock absorbers in the suspension system of a car are in such bad shape that they have no effect on the behavior of the springs attached to the axles. Each of the identical springs attached to the front axle supports 351 kg. A person pushes down on the middle of the front end of the car and notices that it vibrates through four cycles in 2.9 s. Find the spring constant of either spring.
To find the spring constant of the spring, we need to 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.
Hooke's Law can be expressed as:
F = -k * x
Where:
F is the force exerted by the spring (in Newtons)
k is the spring constant (in Newtons per meter)
x is the displacement of the spring from its equilibrium position (in meters)
In this problem, we are given that each spring supports a weight of 351 kg. We can calculate the force exerted by the spring using the formula:
F = mg
Where:
m is the mass (in kilograms)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Substituting the values, we have:
F = 351 kg * 9.8 m/s^2
Now, let's determine the displacement of the springs. We are given that the front end of the car vibrates through four cycles in 2.9 seconds. Each cycle corresponds to one complete oscillation or two displacements. Thus, the number of displacements is 4 cycles * 2 = 8 displacements.
The time it takes for one complete oscillation is given by:
T = t / n
Where:
T is the period (in seconds)
t is the total time (in seconds)
n is the number of oscillations
Substituting the values, we have:
T = 2.9 s / 8
Now, we can determine the displacement of the springs using the formula:
x = A * sin(2πt / T)
Where:
x is the displacement (in meters)
A is the amplitude (maximum displacement)
t is the time (in seconds)
T is the period (in seconds)
Since we are not given the amplitude, we will assume it to be the displacement at the equilibrium position, which is zero. Therefore, we have:
x = A * sin(2πt / T) = 0
Since the displacement is zero, we can assume that at t = 0, the spring is at its equilibrium position. Thus, the initial phase angle is zero, and the equation becomes:
A * sin(0) = 0
This means that the value of sine at 0 is 0, which implies that A * 0 = 0. Therefore, the displacement is zero.
Now that we know the displacement is zero, we can use Hooke's Law to find the spring constant:
F = -k * x
Since x = 0, the force exerted by the spring is also zero. This means:
0 = -k * 0
Since any number multiplied by zero is zero, we can conclude that the value of the spring constant is also zero.
Therefore, the spring constant of either spring is 0 N/m.