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pg 44 - -n:o Imwpmms.fi K 4:mnal r_c.v ‘ 44 Chapter 3...

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Unformatted text preview: -n:o Imwpmms; ._ .. . ._,__._...-..fi..-.__ K 4. :mnal' r._..-_c...v ‘ 44 Chapter 3 Motion in Two and Three Dimensions Find the orbital period (the time to complete one orbit) of a space shut- tle. in circular orbit at an altitude of .250 km, where the acceleration of gravity ’13-‘51th of its surface value. [dreamer This: is a problem about uniform c-imular motion. altitude, not the orbital radius, and We want the period, not the speed. ' so outplan is to write'the‘ speed in terms ofthe'pcriod and use the re- sult'in Equation 3.:1_6. The. crb‘tal altitude-is the distance from Earth’s surface, so we’ll need-to add Earth’s radius to get the orbital radius r. swim-rs The speed i; is-‘tha orbital circumference, 27m divided by the period "I; Using this in Equation 3.16 givas ‘ I (Iltrrri‘T)2 b 4.,117. r T2 v2 ' 161-.th 7' _ Uniform Circular Motion: Engineering a'Road a. engineer is designingaflat, horizontal road for an so kin/h speed limit (that’s 22.2 mls): If the mind urn acceleration of a vehicle on- this road is 1.5 mfsfi what‘s the minimum safe radius for curves in thermal? "ETERPRET Even though acurve is only a portion of a circle, we can " st'dl. interpret this: problem as involving'nniforin circular motion. -: until—93 Equation 3 .16. c: fir; deft-amines the acceleration - given ' the speed and radius: Here website the acceleration and speed, so our - ‘plén: is, to solve for the farms Uniform Circular Motion: Calculating a Spacé'Shuttlé.Orbit. Appendix 5 lists Earth’s radiusasRE = 6.37 Mm giving an orhital ra- dius r e. R3 _+ 259 km : 6.62 Mm. Solving our acoelc'rat'ion expression for the period then giVes T i V 47am: 5355 3': 89 min, where We used a}: mag; ' , “ . ' assess“ Make sense?-You’ve_ probably heard thatzastronauts orbit Earth in about an hour and a half,_expcriflncing multiple sunrises and sunsets in a 24-hour day. Our answer of 89 ruin is certainly consistent with that. There’s no choice here; for a given orbital radius,.Earth’s size and. mass dete’miine the period. Because astronauts’ orbits are hunted to a few hundred Idiomctcrs, a distance small compared 'with' R5, variaw tionsin g and T are minimal. Any such “low Earth. orbit” has a period of approximately 90. ruin At higher altitudes. the decline in the strength-of gravity‘becomes more noticeable and periodslengthen; the Moon-Tor example, orbits in?! days._ We‘ll discuss orbits more in Chapter 8. - _ - - . ' l warrants Using the given numbers, we - have r=vi2ln : (mam/simjmsi = 329' m. - mess-s Make sense? Aspee'd of 80' km/h is pretty fast,_ so we need a. wide curve to keep the required acceleration below its design value, if _' the curve is sharper,.'yehiclcs may slide off the read. We‘ll see'mOre clearly: in subsequent. chapters how vehicles manage tonegotiatc higha - speedcnrvca . Nonuniform Circular Motion The car kw, so its tangentia acceleration E, is oppositeits velocity. nitride is still vzlr, 516133.22 Acceleration of a car that slows as it rounds a curve. What if an. object moves in a circular path but its speed changes? Then it has components of acceleration both perpendicular and parallel to its velocity. The former, the radial ac- celeration an is what changes the direction to keep the object in circular motion. Its mag- With 1) now the instantaneous speed. The parallel component of acceleration; also called tangential acceleration 0, because it’s tangent to the circle, changes the speed but not the direction. Its magnitude is therefore die rate of change of The radial acceleration a, changes only the Speed, or dv/dt. Figure 3.22 shows these two acceleration components for a car rounding direction of motion. a curve Finally, what if the radius of a curved path changes? At any point on a curve we can de— fine a radius of curvature. Then the radial acceleration is still v2/r, and it can vary if either v or r changes along the curve. The tangential acceleration is still tangent to The curve. and it still describes the rate of change of speed. So it’s straightforward to generalize the ideas of uniform circular motion to cases where the motion is nonuniform either because the speed changes, or because the radius changes, or both. GOT IT? 3.4 The figure shows velocity vectors for four points on a noncircular path. Rank order the centripetal accelerations at these points given v. = v4 and v2 = V}. V3 _ ...
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