A mass of 0.24 kg is attached to a spring and
is set into vibration with a period of 0.15 s.
What is the spring constant of the spring?
Answer in units of N/m
To find the spring constant of the spring, we can use the equation
T = 2π * sqrt(m/k)
where T is the period, m is the mass, and k is the spring constant.
Given:
m = 0.24 kg
T = 0.15 s
Plugging in these values into the equation, we have:
0.15 s = 2π * sqrt(0.24 kg / k)
To isolate the spring constant (k), we need to square both sides of the equation:
(0.15 s)^2 = (2π)^2 * (0.24 kg / k)
0.0225 s^2 = 4π^2 * (0.24 kg / k)
Next, we isolate the (0.24 kg / k) term on one side:
(0.24 kg / k) = 0.0225 s^2 / 4π^2
Now, we can solve for k by taking the reciprocal of both sides:
k = (0.24 kg) / (0.0225 s^2 / 4π^2)
Evaluating this expression, we have:
k ≈ 4π^2 * (0.24 kg) / 0.0225 s^2
k ≈ 4π^2 * 0.24 kg / 0.0225 s^2
k ≈ 50,265 N/m
Therefore, the spring constant is approximately 50,265 N/m.
To find the spring constant of the spring, we can 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 represented by the formula:
F = -k * x
where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, we are given the period T of the vibration, which is the time it takes for the mass to complete one full cycle. The period T is related to the frequency f of the vibration by the equation:
T = 1 / f.
We can also calculate the frequency f using the formula:
f = 1 / T.
Given that T = 0.15 s, we can calculate the frequency as:
f = 1 / 0.15 = 6.67 Hz.
The frequency is related to the angular frequency ω by the equation:
ω = 2πf.
Plugging in the value of f, we get:
ω = 2π * 6.67 = 41.89 rad/s.
The angular frequency ω is related to the spring constant k by the equation:
ω = √(k/m),
where m is the mass attached to the spring.
Given that the mass m = 0.24 kg, we can rearrange the equation to solve for the spring constant k:
k = ω^2 * m.
Plugging in the values of ω and m, we have:
k = (41.89)^2 * 0.24 = 215.39 N/m.
Therefore, the spring constant of the spring is 215.39 N/m.
P = 2*pi*sqrt(m/k) = 0.15 s
m = 0.24 kg
Solve for k.