An astronaut measures her mass by means of a device consisting of a chair attached to a large spring. The period of oscillation she undergoes when set into motion is used to determine her mass. If her mass, together with the chair, is 172kg, and the time required for her to undergo 10 full oscillations is 27 seconds, what is the spring constant?
T=2π•sqrt{m/k}
T=t/N
t/N=2π•sqrt{m/k}
(t/2πN)²=m/k
k=m(2πN/t)²
To find the spring constant, we can use the formula for the period of oscillation of a mass-spring system:
T = 2π√(m/k)
Where:
T = Period of oscillation
m = Mass (including the chair)
k = Spring constant
Given:
Mass, m = 172 kg
Number of oscillations, N = 10
Time for N oscillations, t = 27 seconds
First, we need to find the period of oscillation for one oscillation, which can be calculated using the formula:
T = t / N
T = 27 s / 10 = 2.7 s
Now, we can rearrange the formula for the period of oscillation to solve for the spring constant:
k = (4π²m) / T²
k = (4π² * 172 kg) / (2.7 s)²
k = (4 * 3.14² * 172 kg) / (2.7 s)²
k ≈ 1005.98 N/m
Therefore, the spring constant is approximately 1005.98 N/m.
To find the spring constant, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement of the spring. Mathematically, Hooke's law can be expressed as:
F = -kx
Where:
F is the force applied by the spring,
k is the spring constant, and
x is the displacement from the equilibrium position.
In this case, the period of oscillation (T) is given, which is the time required for the astronaut to undergo one full oscillation. The period (T) is related to the angular frequency (ω) of the oscillation by the equation:
T = 2π/ω
We can use this relationship to find ω.
Given:
Mass of the astronaut and chair (m) = 172 kg
Number of full oscillations (n) = 10
Time for n full oscillations (t) = 27 seconds
First, let's find the period (T) using the given time (t) and number of oscillations (n):
T = t / n
T = 27 seconds / 10
T = 2.7 seconds
Next, let's find the angular frequency (ω) using the period (T):
ω = 2π / T
ω = 2π / 2.7 seconds
ω ≈ 2.322 rad/s
Now, let's find the force (F) exerted by the spring using the mass (m) and the acceleration due to gravity (g = 9.8 m/s²):
F = mg
F = 172 kg * 9.8 m/s²
F ≈ 1685.6 N
Lastly, let's find the spring constant (k) using Hooke's law:
F = -kx
Since the force (F) is balanced by the gravitational force (mg), we have:
mg = -kx
Rearranging the equation, we get:
k = -mg / x
To find x, which is the maximum displacement or amplitude, we can use the relationship between the maximum displacement (A) and the period (T):
A = (2π / ω) * √(mg / k)
Since the maximum displacement (amplitude) occurs when F = -kx, we can substitute F = mg and solve for x:
mg = -kx
x = -mg / k
Now we can substitute the values into the equation to find x:
A = (2π / ω) * √(mg / k)
A = (2π / ω) * √((-mg / k) * (mg / k)) [Substituting x = -mg / k]
A = (2π / ω) * √((m²g²) / (k²))
Simplifying further:
A = (2π / ω) * (mg / k)
A = (2π / ω) * (172 kg * 9.8 m/s²) / k
Now, substitute the values of ω, m, and g to find x:
2.322 = (2π / 2.322) * (172 kg * 9.8 m/s²) / k
Now, we can solve for k:
k = (2π / 2.322) * (172 kg * 9.8 m/s²) / 2.322
After evaluating this equation, we find the spring constant value.