A spring with a spring constant of 1.33 102 N/m is attached to a 1.6 kg mass and then set in motion.
(a) What is the period of the mass-spring system?
(b) What is the frequency of the vibration?
To calculate the period and frequency of a mass-spring system, we can use the following formulas:
Period (T) = 2π√(m/k)
Frequency (f) = 1/T
Given:
Spring constant (k) = 1.33 × 10^2 N/m
Mass (m) = 1.6 kg
(a) To find the period of the mass-spring system, we need to solve for T using the formula:
T = 2π√(m/k)
T = 2π√(1.6 kg / 1.33 × 10^2 N/m)
= 2π√(1.6 kg / 133 N/m)
= 2π√(0.0120 kg/N)
= 2π × 0.1098 s
≈ 0.689 s
Therefore, the period of the mass-spring system is approximately 0.689 seconds.
(b) To find the frequency of the vibration, we can use the formula:
f = 1/T
f = 1 / 0.689 s
≈ 1.45 Hz
Therefore, the frequency of the vibration is approximately 1.45 Hz.
To find the period of the mass-spring system, we can use the formula:
T = 2π√(m/k)
where T is the period, m is the mass of the object, and k is the spring constant.
(a) Let's substitute the given values into the formula:
T = 2π√(1.6 kg / 1.33 * 10^2 N/m)
Calculating the square root part first, we get:
T = 2π√(1.6 kg / 133 N/m)
Now, divide the mass by the spring constant:
T = 2π√(0.01203 s^2)
Finally, calculate the square root:
T ≈ 2π * 0.1098 s
Using a calculator, we can find that:
T ≈ 0.6882 s
Therefore, the period of the mass-spring system is approximately 0.6882 seconds.
(b) The frequency of the vibration is the reciprocal of the period, so:
f = 1/T
Substituting the value of T we calculated in part (a):
f = 1/0.6882 s
Calculating this, we find that:
f ≈ 1.453 Hz
Therefore, the frequency of the vibration is approximately 1.453 Hertz.