The amplitude of a system moving in simple harmonic motion is doubled. Determine by what factor does period change
since T=2pi(m/k)^(1/2), so it doesnt change. When i typed in unchanged, it says wrong since "the correct answer is not zero". Should i put in "1" then?
Yes, "1" would be the factor, since the period does not change.
No, the correct answer is not zero or 1. Let's break it down step by step to determine the correct factor by which the period changes when the amplitude is doubled.
The period of motion for a system in simple harmonic motion is given by the formula:
T = 2π√(m/k)
where T is the period, m is the mass of the object, and k is the spring constant.
In this case, we are doubling the amplitude. The amplitude of motion is directly related to the maximum displacement of the system from its equilibrium position. Therefore, doubling the amplitude means that the maximum displacement has also doubled.
Now, let's consider the relationship between the amplitude and the period. The period is the time taken for one complete oscillation, which means it is the time taken for the system to go from its maximum displacement in one direction to its maximum displacement in the opposite direction.
When the amplitude is doubled, the maximum displacement is now twice as much. This means that the system will take longer to go from one extreme to the other. Consequently, the period will increase.
To determine the factor by which the period changes, we need to find the ratio of the new period to the original period:
(New Period) / (Original Period)
Since the period is directly proportional to the square root of the mass divided by the spring constant, we can rewrite this ratio as:
√(New mass / New spring constant) / √(Original mass / Original spring constant)
However, we only doubled the amplitude, which means the mass and the spring constant remain the same. Therefore, the ratio becomes:
√(Original mass / Original spring constant) / √(Original mass / Original spring constant)
This simplifies to:
(Original mass / Original spring constant) / (Original mass / Original spring constant)
This ratio is equal to 1, which means that the factor by which the period changes is indeed 1. Thus, the correct answer is 1.