A jellyfish can propel itself with jets of water pushed out of its bell, a flexible structure on top of its body. The elastic bell and the water it contains function as a mass-spring system, greatly increasing efficiency. Normally, the jellyfish emits one jet right after the other, but we can get some insight into the jet system by looking at a single jet thrust. Figure shows a graph of the motion of one point in the wall of the bell for such a single jet; this is the pattern of a damped oscillation. The spring constant for the bell can be estimated to be 1.2 N/m.
A) What is the period for the oscillation?
B) Estimate the effective mass participating in the oscillation. This is the mass of the bell itself plus the mass of the water.
C) Consider the peaks of positive displacement in the graph. By what factor does the amplitude decrease over one period? Given this, what is the time constant for the damping?
A) To find the period of the oscillation, we need to determine the time it takes for the motion to complete one full cycle. In the graph, the period corresponds to the time between two consecutive peaks (or valleys) of the oscillation.
To find the period, we can measure the time difference between two adjacent peaks. Let's denote this time difference as T. Typically, graphs like these have a horizontal axis representing time.
B) The effective mass participating in the oscillation includes both the mass of the bell and the mass of the water contained within it. To estimate the effective mass, we need to determine the total mass of the bell and the water.
We can use the gravitational force equation: F = m*g, where F is the weight of the object, m is the mass, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
First, let's determine the weight of the bell, which is given by the equation F_bell = m_bell * g. We can rearrange this equation to find the mass of the bell: m_bell = F_bell / g.
Next, we need to determine the weight of the water. Since the water is contained within the bell, it experiences the same force. We can use the equation F_water = m_water * g and rearrange it to find the mass of the water: m_water = F_water / g.
Finally, the effective mass participating in the oscillation is the sum of the bell's mass and the water's mass: m_effective = m_bell + m_water.
C) Now, let's consider the peaks of positive displacement in the graph. The amplitude represents the maximum displacement from the equilibrium position. By observing the graph, we can measure the initial peak amplitude and the amplitude of the subsequent peaks over one period.
To find the factor by which the amplitude decreases, we divide the amplitude of the subsequent peak by the initial peak amplitude. Let's call this factor "A."
Given this amplitude decrease factor, we can find the time constant for damping, often represented by the symbol "τ." The time constant is related to the decay of the oscillation and can be calculated using the formula τ = T / ln(A), where T is the period and ln(A) is the natural logarithm of the amplitude decrease factor.
Using these methods, we can find the answers to each part of the question.