(a) Determine a differential equation for the velocity v(t) of a mass m sinking in water that...
Question:
(a) Determine a differential equation for the velocity v(t) of a mass m sinking in water that imparts a resistance proportional to the square of the instantaneous velocity and also exerts an upward buoyant force whose magnitude is given by Archimedes principle. See Problem 18 in Exercises 1.3. Assume that the positive direction is downward.
(b) Solve the differential equation in part (a).
(c) Determine the limiting, or terminal, velocity of the sinking mass.
Data from problem 18
A cylindrical barrel s feet in diameter of weight w lb is floating in water as shown in figure (a). After an initial depression the barrel exhibits an up-and-down bobbing motion along a vertical line. Using figure (b), determine a differential equation for the vertical displacement y(t) if the origin is taken to be on the vertical axis at the surface of the water when the barrel is at rest. Use Archimedes principle: Buoyancy, or upward force of the water on the barrel, is equal to the weight of the water displaced. Assume that the downward direction is positive, that the weight density of water is 62.4 lb/ft3, and that there is no resistance between the barrel and the water.
Transcribed Image Text:
s/2 |s/2 surface }y(t) (b) (a)
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a Let be the weight density of the water and V the volume of the object Archimedes princip...View the full answer
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Density helps us predict whether something will float. Density is super important to consider when building things like ships and submarines. The experiment in the video examines the water density by comparing a glass of water containing sugar with a simple glass of water