How you find the oscillation period of a mass if not given the mass, or spring constant?

A mass is attached to a vertical spring, which then goes into oscillation. At the high point of the oscillation, the spring is in the original unstretched equilibrium position it had before the mass was attached; the low point is 3.7cm below this. Find the oscillation period, T.

To find the oscillation period (T) of a mass attached to a vertical spring, we can use the relationship between the period, mass (m), and the spring constant (k). However, in this particular question, the mass and spring constant are not given.

To solve this problem, we can use the fact that the high point (where the spring is in the original unstretched equilibrium position) and the low point (3.7cm below the high point) are symmetric points in the oscillation.

Here's how we can find the oscillation period:

1. Determine the amplitude (A):
- The high point is the amplitude of the oscillation because it represents the maximum distance from the equilibrium position.
- In this case, the high point is the original unstretched equilibrium position, so the amplitude (A) is 0.

2. Determine the displacement (x):
- The displacement is the distance from the equilibrium position at any point during the oscillation.
- In this case, the low point is 3.7cm below the high point. Therefore, the displacement (x) is -3.7cm (negative because it's below the equilibrium position).

3. Determine the time period (T):
- The time period is the time it takes for the mass to complete one full oscillation (i.e., go from one extreme point to another and back).
- Since the high point and low point are symmetric, it means that the mass will take the same amount of time to move from the high point to the equilibrium position as it takes to move from the equilibrium position to the low point.
- Therefore, we can calculate the total time taken for one complete oscillation as twice the time taken to go from the equilibrium to the low point.
- The time taken for this part of the oscillation can be calculated using the equation for simple harmonic motion:
T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant.
- As we don't have the mass or the spring constant, we can ignore them because we want to find T without knowing those values.

4. Calculate T:
- Since both the mass and spring constant are unknown and not necessary for finding T in this case, we can directly calculate the time period using the displacement:
T = 2π√(x/g), where T is the period, x is the displacement, and g is the acceleration due to gravity (assuming the oscillation happens on Earth).
- Substituting the displacement (-3.7cm or -0.037m) into the equation gives us:
T = 2π√(-0.037/9.8)

5. Solve for T:
- Calculate the value of the expression 2π√(-0.037/9.8) using a calculator or math software.
- The result will give you the period (T) of the oscillation in seconds.

Note: It's worth mentioning that without knowing the mass and spring constant, it is not possible to determine the exact values of the period or any other specific properties of the system. However, using the displacement and acceleration due to gravity, we can still calculate an approximate value for the period.