A 210 g air-track glider is attached to a spring. The glider is pushed in 12.0 cm against the spring, then released. A student with a stopwatch finds that 13 oscillations take 16.0 s. Find the spring constant k.
T = 16 s/ 13 = 1.23 s
T = 2pi sq rt (m/k)
1.23 = 2 pi sq rt (.21 g/k)
k = 5.473
Well, it seems like that student with a stopwatch might have missed their calling as a comedian! But let me help you with the calculation.
We can start by finding the period of oscillation (T) using the given values:
Period (T) = Total time (t) / Number of oscillations (n)
= 16.0 s / 13
Now, since frequency (f) is the reciprocal of the period (T), we can find the frequency of oscillation:
Frequency (f) = 1 / T
Once we have the frequency, we can find the angular frequency (ω) using the formula:
Angular Frequency (ω) = 2πf
Now, the formula connecting angular frequency (ω) and the spring constant (k) is:
ω = √(k / m),
where m is the mass of the glider.
Since we know the angular frequency (ω) and the mass (m), we can rearrange the formula to solve for the spring constant (k):
k = ω^2 * m
And that's it! Plug in the values and calculate the spring constant (k). Just don't expect the spring constant to solve any more jokes for you!
To find the spring constant (k), we can use the formula for the period of a mass-spring system:
T = 2π * sqrt(m / k)
Where:
T = period of oscillation
m = mass of the glider
k = spring constant
We are given the period (T) and the mass (m), so we can rearrange the formula to solve for k.
First, let's calculate the time period per oscillation:
Time period per oscillation (T') = Total time / Number of oscillations
T' = 16.0 s / 13
T' ≈ 1.23 s
Next, we can solve the rearranged formula for k:
k = (4π^2 * m) / T^2
Substituting the given values:
k = (4π^2 * 0.210 kg) / (1.23 s)^2
Calculating this expression will give us the value of the spring constant, k.
To find the spring constant (k) in this problem, we can use the formula for the period (T) of an oscillating mass-spring system:
T = 2π√(m/k),
where T is the period, m is the mass of the glider, and k is the spring constant.
Let's start by finding the period (T) of a single oscillation. We are given that the glider takes 16.0 s to complete 13 oscillations. We can divide the total time by the number of oscillations to find the period of a single oscillation:
T = 16.0 s / 13
T ≈ 1.23 s
Now, we can rearrange the formula for T to solve for the spring constant (k):
T = 2π√(m/k)
Squaring both sides of the equation:
T^2 = 4π^2(m/k)
Rearranging again to isolate k:
k = 4π^2m / T^2
Now, substitute the given values:
m = 210 g = 0.210 kg
T = 1.23 s
k = 4π^2(0.210 kg) / (1.23 s)^2
Using a calculator, we can evaluate this expression:
k ≈ 10.83 N/m
Therefore, the spring constant (k) is approximately 10.83 N/m.
see link 1 below under related questions.
a very similar problem was answered.