A block is connected between two springs on a frictionless horizontally level track. The other end of each spring is respectively anchored to a vertical pin set at opposite ends of the track. The spring constant for one spring is k = 100N/m and the mass of the block is 0.300 kg. If the period of oscillation T is 0.21 s, what is the spring constant of the other spring?
Find the series k from the mass and period of oscillation. w= 2PIf=2PI/T
w= sqrt (k/m)
Then, having that k
1/k= 1/k1 + 1/k2
solve for k2
To find the spring constant of the second spring (k2), we can follow these steps:
1. Start with the angular frequency formula for simple harmonic motion: ω = √(k/m), where ω is the angular frequency, k is the spring constant, and m is the mass of the block. Rearrange the formula to solve for k: k = mω^2.
2. Use the formula for the angular frequency in terms of the period of oscillation: ω = 2π/T, where T is the period of oscillation.
3. Substitute the value of angular frequency into the equation for the spring constant: k = m(2π/T)^2.
4. Now, we have the spring constant for the first spring (k1), which is given as k1 = 100 N/m. We need to find the spring constant for the second spring (k2). Apply the series combination formula for two springs in parallel: 1/k = 1/k1 + 1/k2.
5. Rearrange the formula to solve for k2: 1/k2 = 1/k - 1/k1.
6. Substitute the values of k and k1 into the equation: 1/k2 = 1/(m(2π/T)^2) - 1/k1.
7. Calculate the reciprocal of both sides: k2 = 1/(1/(m(2π/T)^2) - 1/k1).
8. Simplify the expression and compute the value of k2.