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 oscillation, we need to determine the time it takes for one complete cycle. In the given graph, the oscillation goes from the maximum positive displacement, back to zero, then to the maximum negative displacement, and finally returns to zero again. This constitutes one full cycle.

From the graph, you can measure the time it takes to complete one cycle by observing the time interval between two consecutive peaks or troughs. Once you have this time interval, you can calculate the period using the formula:

Period (T) = Time interval between two consecutive peaks/troughs

B) The effective mass participating in the oscillation consists of the mass of the bell itself and the mass of the water inside it. To estimate this, you can use the formula:

Effective mass (m) = m_bell + m_water

Here, m_bell represents the mass of the bell and can be determined by weighing it. For the mass of water, you can measure the volume of water displaced by the oscillating bell and then multiply it by the density of water (approximately 1000 kg/m^3) to obtain the mass.

C) The amplitude of the oscillation decreases over one period. To determine the factor by which the amplitude decreases, you can compare the amplitude of the first peak with the amplitude of the next peak in the positive displacement region.

Let's say the amplitude of the first peak is A0, and the amplitude of the next peak is A1. Then, the factor by which the amplitude decreases is given by:

Decrease factor = A1/A0

To find the time constant for damping, we need to know the decrease factor (A1/A0). The time constant (τ) can be calculated using the formula:

Time constant (τ) = Period (T) / ln(Decrease factor)

By substituting the previously calculated period and decrease factor values into this formula, you can find the time constant for damping.