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 303 kg. A person pushes down on the middle of the front end of the car and notices that it vibrates through six cycles in 2.9 s. Find the spring constant of either spring
T=t/N =2.9/6 = 0.48 s
T=2π•sqrt(m/k) = >
k=4 π²m/T²= …
To find the spring constant of either spring, we can start by calculating the frequency of the vibration.
Frequency (f) is calculated as the number of cycles (n) divided by the time taken (t):
f = n / t
In this case, the car vibrates through six cycles in 2.9 seconds, so:
f = 6 / 2.9
Now, we can calculate the angular frequency (ω) using the formula:
ω = 2πf
Plugging in the value of frequency calculated earlier:
ω = 2π * (6 / 2.9)
Next, we can calculate the mass supported by each spring. As given in the question, each spring supports 303 kg, and there are two identical springs:
Mass (m) = 303 kg / 2
Finally, we can use the formula to find the spring constant (k):
k = m * ω^2
Plugging in the values calculated earlier:
k = (303 kg / 2) * (2π * (6 / 2.9))^2
Note: Make sure all the units are consistent in the calculations.
To find the spring constant of either spring, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement of the spring from its equilibrium position. The formula for Hooke's Law is:
F = -k * x
Where:
F is the force exerted by the spring,
k is the spring constant, and
x is the displacement of the spring.
In this case, the force exerted by the spring is the weight being supported by the spring, and the displacement is the amplitude of the vibration.
We can start by finding the weight being supported by each front spring. Given that each spring supports 303 kg, the weight is equal to the mass multiplied by the acceleration due to gravity (9.8 m/s^2):
Weight = mass * g
Weight = 303 kg * 9.8 m/s^2
Weight = 2969.4 N
Since the springs are identical, the force exerted by each spring is the same. Therefore, the equation becomes:
2969.4 N = -k * x
Next, we need to find the displacement, x. The question states that the car vibrates through six cycles in 2.9 seconds. One complete cycle includes both the upward and downward motions of the vibration, so the number of cycles corresponds to the number of times the spring goes from its highest point to its lowest point. Since there are six cycles, the spring goes through three complete oscillations:
Number of oscillations = 6 cycles / 2 = 3 oscillations
The displacement, x, is equal to twice the amplitude of the vibration. In one complete oscillation, the spring moves from its highest point to its lowest point and back to the highest point, covering the full amplitude twice. Therefore, the total displacement is six times the amplitude (3 oscillations * 2 times):
x = 6 * amplitude
Finally, we can substitute the values into the equation and solve for the spring constant, k:
2969.4 N = -k * 6 * amplitude
The negative sign indicates that the force of the spring is in the opposite direction of the displacement. Solving for k:
k = -(2969.4 N) / (6 * amplitude)