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Transcript
METR 402: Intro Atmos & Ocn Dynamics Dr. Dave Dempsey Spring 2002 Solutions to Quiz #5 (1) [6 pts] For each of the real and apparent forces/mass listed below, state what you can in general terms about: I. its magnitude; and II. its direction. (a) the pressure-gradient force/mass Magnitude: Proportional to the magnitude of the pressure gradient, and inversely proportional to the density. Direction: Parallel to the pressure gradient, toward lower pressure. (b) true gravity/mass (sometimes called “gravitation”) Magnitude: Proportional to the mass of the earth; inversely proportional to the square of the distance between the center of mass of the earth and the center of mass of the object in question. Direction: Toward the center of mass of the earth. (c) friction (direction only) Direction: Opposite the direction of motion of the object on which it acts. (There are exceptions to this, though!) (d) centrifugal force/mass when rotating with the earth. Magnitude: Proportional to the distance from the axis of rotation and to the square of the angular speed of the earth. Direction: Outward from and perpendicular to the earth’s axis of rotation. (e) effective force/mass of gravity Magnitude: Equal to (at the poles) or slightly less than true gravity (minimum at the equator). Direction: Perpendicular to the earth’s surface, “downward”. (f) the Coriolis force/mass Magnitude: Proportional to the earth’s angular speed and to the component of an object’s velocity that is perpendicular to the earth’s axis of rotation. Direction: Perpendicular to the axis of rotation and to an object’s velocity relative to the earth. Answer your choice of only one of the following three questions: (2) [4 pts] Explain why the rotating earth cannot be a perfect sphere but rather must be slightly flattened at the poles and slightly bulging at the equator. (Adopt an inertial frame of reference and invoke Newton’s Second Law appropriately. A diagram will help.) Pieces of the earth exhibit perfect circular motion, which means they are accelerating directly toward the earth’s axis of rotation at a rate proportional to their distance from the axis and to the square of the earth’s angular speed. The only forces acting on pieces of the earth to produce this acceleration are true gravity (gravitation), which pulls toward the center of mass of the earth (not the axis of rotation); and pressure forces exerted by surrounding pieces of the earth. The net effect of the pressure forces is to push in a direction perpendicular to the earth’s surface, “upward”. If the earth were a sphere, there is no way that these two forces could combine to produce a net force (and hence an acceleration) toward the axis of rotation. The only way that these two forces can combine to produce the required acceleration is for the earth to be flattened at the poles and bulge at the equator, to form an oblate spheroid. Then the normal force can cancel the part of gravity normal to the earth’s surface and leave the remainder of true gravity to pull pieces of the earth toward the axis of rotation, maintaining perfect circular motion. (3) [4 pts] For a given pressure gradient, the pressure-gradient force/mass increases with decreasing fluid density. Explain why, physically. Also, for a given fluid density, the pressure-gradient force/mass increases with increasing pressure gradient. Explain why, physically. The pressure-gradient force per unit mass consists of the net force due to pressure on a fluid parcel (or any object), divided by its mass. The net force due to pressure on the object of given size depends on the pressure difference between opposite sides of the object and the surface area of the object (greater pressure difference and greater surface area mean greater net force). For an object with lower density but the same size, the net pressure force will be the same but it’s mass is less, so the net force per unit mass will be greater. The pressure gradient is a measure of how rapidly the pressure varies from place to place. For an object of given size, the larger the pressure gradient the more rapidly pressure will vary from one side of the object to the other. This implies a larger pressure difference from one side to the other, and hence a larger net force on the object, and hence a larger pressure gradient force per unit mass. (4) [4 pts] Even in the absence of any torque, a fluid parcel that moves northward in the Northern Hemisphere experiences an acceleration eastward relative to the earth’s surface. Explain why, in terms of the conservation of (absolute) angular momentum. Conservation of absolute angular momentum requires that the product of an object’s distance from the axis of rotation, R, and its absolute tangential speed, Va, is conserved (that is, RVa doesn’t change following the object). In our context, the object’s absolute tangential velocity is the sum of the earth’s speed Ve = R and the tangential (east/west) component of the object’s velocity relative to the earth, u (so Va = R in the Northern Hemisphere is moving closer to the axis of rotation, which requires that it’s absolute tangential speed, Va = R speed of the earth actually decreases, so u must become more positive or less negative (that is, the object must accelerate eastward) enough so that R change in u occurs in the absence of any real forces acting in that direction (that is, tangentially), consistent with the idea that the Coriolis force is an apparent force, not a real one. (5) [4 pts] In a frame of reference rotating with the earth, an object that moves eastward in the Northern Hemisphere experiences an acceleration southward and upward. Explain why. In a frame of reference rotating with the earth, we observe an object at rest relative to the earth not to be accelerating and hence must experience no net force acting on it. In particular, true gravity/mass, the normal (vertical) force/mass (a pressure-gradient force/mass), and an apparent centrifugal force/mass = R = (Ve/R)R, must sum to zero. However, an object that moves eastward relative to the earth will experience a centrifugal force = ((Ve+u)/R)R which is greater than that needed to balance gravity and the normal force. The extra centrifugal force will make the object appear to accelerate outward, away from the axis of rotation. In the Northern Hemisphere, this direction has components southward and upward.