A mass on a string of unknown length oscillates witha period of 6.0s
What is the period if the mass is doubled?
T= 2PI sqrt (m/k) Double mass, what is the sqrt 2?
To find the period if the mass is doubled, we can use Newton's law of motion for simple harmonic motion. The period (T) of an object in simple harmonic motion is defined as the time taken for one complete cycle of motion. It can be calculated using the equation:
T = 2π√(m/k)
where:
T is the period of oscillation,
m is the mass of the object, and
k is the spring constant.
In this case, the object is a mass on a string, where the string is acting as a spring. The period is given as 6.0 seconds. However, the length of the string is unknown, and we need to find the period if the mass is doubled.
Let's denote the original mass as "m" and the original length of the string as "L". We can solve for the spring constant (k) using the formula:
k = 4π²m/L²
Since the mass is doubled, the new mass (m') will be 2m. Substituting this into the spring constant equation, we get:
k' = 4π²(2m)/L²
k' = (8π²m)/L²
Now, we can use the new spring constant to find the new period (T'). Substituting the new mass and the new spring constant into the period equation, we have:
T' = 2π√(2m/k')
T' = 2π√(2m)/√((8π²m)/L²)
T' = 2π√(2m)√(L²/(8π²m))
T' = 2π√(2L²/(8π²))
T' = 2πL/√(8π²)
T' = (2π√2)L/(2π)
T' = √2L
Therefore, if the mass is doubled, the new period (T') will be equal to √2 times the original length of the string (L).