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 either spring, we can use the formula for the period of oscillation:
T = 2π * sqrt(m/k)
where:
T is the period of oscillation (in seconds)
π is a mathematical constant approximately equal to 3.14
m is the mass being supported by the spring (in kg)
k is the spring constant (in N/m)
From the given information, we know that the car vibrates through four cycles in 2.9 seconds. The number of cycles is directly related to the period of oscillation (T), so we can determine T by dividing the total time (2.9 s) by the number of cycles (4):
T = (2.9 s) / (4 cycles)
T = 0.725 s/cycle
Now, we need to determine the mass (m) being supported by the spring. Since each spring supports 351 kg, the total mass supported by both springs is 2 * 351 kg = 702 kg. However, since we are finding the spring constant of either spring, we need to divide the total mass by the number of springs, which is 2:
m = (702 kg) / (2 springs)
m = 351 kg
Now we have the values for T and m, so we can rearrange the formula to solve for the spring constant (k):
k = (4π^2 * m) / T^2
Substituting the values:
k = (4 * 3.14^2 * 351 kg) / (0.725 s/cycle)^2
k ≈ 5108 N/m
Therefore, the spring constant of either spring is approximately 5108 N/m.