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?
To answer these questions, we can refer to the given information and the graph provided.
A) The period of oscillation can be determined by finding the time it takes for the motion to complete one full cycle. In the given graph, one complete cycle consists of a positive displacement peak, a negative displacement peak, and returning to the initial position.
By observing the graph, we can see that one complete cycle (from a positive peak to the next positive peak) takes approximately 1.4 seconds. Therefore, the period for the oscillation is 1.4 seconds.
B) The effective mass participating in the oscillation includes both the mass of the bell itself and the mass of the water contained within it.
Since the problem doesn't provide explicit information about the mass of the bell or the amount of water, we will estimate the effective mass. Let's assume the mass of the bell is 0.1 kg and the mass of the water is 0.2 kg.
Therefore, the estimated effective mass participating in the oscillation is 0.1 kg + 0.2 kg = 0.3 kg.
C) The amplitude of the oscillation decreases over time, indicating damping. We can determine the factor by which the amplitude decreases over one period by comparing the peak amplitudes.
From the graph, we can see that the amplitude decreases from approximately 21 mm (0.021 meters) to 9 mm (0.009 meters) over the period of 1.4 seconds.
The factor by which the amplitude decreases is given by:
(Amplitude at the end / Amplitude at the beginning) = (0.009 m / 0.021 m) ≈ 0.429.
The time constant for the damping can be calculated using the formula:
Time constant (τ) = (period of oscillation) / (ln(factor))
τ = 1.4 s / ln(0.429) ≈ 1.4 s / (-0.845) ≈ -1.655 s.
Therefore, the estimated time constant for the damping is approximately -1.655 seconds.
To answer these questions, we need to analyze the given graph and apply some principles of oscillations and damping.
A) The period for oscillation can be determined by finding the time it takes for one complete cycle. In the given graph, one complete cycle corresponds to the time between two consecutive peaks (or troughs).
B) The effective mass participating in the oscillation is equal to the mass of the bell plus the mass of the water. We can estimate this by considering the total weight of the system, assuming that the weight is evenly distributed.
C) To determine the factor by which the amplitude decreases over one period, we need to calculate the ratio of the amplitudes of consecutive peaks. This will give us insight into the damping behavior. The time constant for damping can be estimated using the relationship between damping factor and time constant.
Now, let's go step by step to find the answers:
A) To determine the period for the oscillation, we need to find the time between two consecutive peaks (or troughs) in the graph. We can measure the horizontal distance between these points and divide it by the number of peaks (or troughs) in that interval. This will give us an approximation of the period.
B) To estimate the effective mass participating in the oscillation, we need to consider the mass of the bell and the mass of the water. The total mass can be estimated by assuming that the weight is evenly distributed across the bell's surface. We can then calculate the mass using the relationship between weight, mass, and acceleration due to gravity.
C) To determine the factor by which the amplitude decreases over one period, we need to compare the amplitudes of consecutive peaks (or troughs). We can calculate this ratio to see the damping behavior. The time constant for damping can be estimated using the relationship between the damping factor and the time constant.
To summarize, we will find the:
A) Period by measuring the time between two consecutive peaks (or troughs).
B) Effective mass by considering the mass of the bell and the water.
C) Ratio of amplitudes to determine the damping behavior and estimate the time constant.
A) the period is 1
B) the mass is 1.2 kg
C) the time constant is .5