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A uniform chain hangs in the shape of the catenary
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Mathematical Methods in the Physical Sciences
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- The density of air changes with height. Under some conditions density p, depends on height z, and temperature T according to the following equation where Po and A are both constants. A meteorological balloon ascends (i.e., starts at z = 1 and gains height) over the course of several hours. Complete parts (a) and (b) below. Az P(z,T) = Po e ..... dz v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, (a) Assuming that the balloon ascends at a speed v (i.e., dt as measured by the weather balloon. Choose the correct answer below. dp Az dT O A. dt T2 dt dp %3D dt Az dT dp С. dt %3D + T dt dp Az) dT O D. dt T2) dt (b) Assume that v = 1, Po = 1, and A = 1 and that when t= 0, T= 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = 0, and solve this…arrow_forwardA formula for the drag force is F = dv² (found to be valid for a large regime of data), where F is the drag force, d is a constant (involving the area of the object, density of the fluid, etc.), and v is the relative speed between the object and the fluid. la) Rewrite the drag force equation in the form: lb) In a plot of log F (y- log F = (...) log v + log (.…..) variable) vs log v (x- variable), what do you expect to be the slope value? lb) In a plot of log F (y- variable) vs log v (x- variable), what do you expect to be the y-intercept?arrow_forwardFind the fluid force (in lb) on a circular observation window of radius 1 foot in a vertical wall of a large water-filled tank at a fish hatchery when the center of the window is 8 feet and d feet (d > 1) below the water's surface (see figure). Use trigonometric substitution to evaluate the one integral. Water weighs 62.4 pounds per cubic foot. (Recall that in Section 7.7, in a similar problem, you evaluated one integral by a geometric formula and the other by observing that the integrand was odd.) (a) 8 feet (b) d feet below X lb lb x² + y² = 1 -2 2 Xarrow_forward
- "Hand-in question": While biking, the force of air resistance F (mass times length, per time- squared) appears to be positively related to the speed v (length per time) at which the bike travels and the cross-sectional area A (length-squared) of bike and biker. However, the force involves mass and hence cannot depend on v and A alone. Let the force also depend on the air density p (mass per length-cubed). Hence, the force F can be expressed as some multiplicative combination of v, A and p, that is F = kv" Aª p², for k E R. Find the values of r, y and z that result in the above equation being dimensionally consistent.arrow_forwardIn conclusion, state the relationship between the mass of the spherical object and the distances they were let to roll with reference to the ramp to the knock-back distance of catcher 1. What implications do these relationships give concerning the magnitude of the force which causes the knocking-back of catcher 1?arrow_forwardHow much force must be applied to a 1.8kg ice hockey puck to give it an acceleration of 10 m s-2arrow_forward
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