Unifom Cicula Motion Unifom cicula motion is the motion of an object taveling at a constant(unifom) speed in a cicula path. Fist we must define the angula displacement and angula velocity The angula displacement of a igid object otating about a fixed axis is the angle the object tuns though, : = f - i The convention is a positive displacement is counteclockwise and negative if clockwise. The units ae adians (ad) which ae dimensionless. What is a adian? A adian is the ac length divided by the adius. Since the total ac length (the cicumfeence) is 2π, thee ae 2π adians in 360 o. That woks out to 1 ad = 57.3 o. s = s 2 π (ad) = 360 o If an object is otating about a fixed axis, then in some time t, it sweeps out an angle. As you might expect, the angula velocity is: ω = o t t 0 = t The units of ω ae ad/s. Like with, if the otation is counteclockwise, ω is positive. This is actually the aveage angula velocity. Of couse if we take the limit as t 0 then we have the instantaneous angula velocity We can elate the angula displacement and angula velocity, ω, to the linea (tangential) displacement and velocity. The linea displacement and angula displacement ae elated by the definition of the adian: = s The angula velocity is elated to the tangential velocity by: v = ω Let s wok poblems 6-1, 6-2 and 6-4 as examples
Centipetal Acceleation By the definition, the speed is constant in unifom cicula motion. What about the velocity? The velocity is tangent to the cicle of motion so the diection changes as we go aound the cicle. Thus the velocity changes. Since the velocity is changing, thee must be an acceleation. Conside an object moving aound a cicle. v at t 0 path length v t v at t 1 Since the velocity is tangent to the cicle, the change in the angle of the velocity vecto is as the object moves aound the pat of the cicle defined by. (Think about a little v aow mounted pependicula to a stick. As you move the stick though an angle, the little v aow moves though an angle.) v at t 1 v at t 0 v t v We want to get at the acceleation. Since the angle is the same fo ou velocity vectos and positions, we can ask what happens in the case whee we let t 0. The side of the cicle (the path length cuve in the pevious dawing) becomes a staight line. We have two simila tiangles so: v = v t v a c = v t = v2 In tems of the angula velocity (ω) the magnitude of the acceleation is: a c = ω 2
What about the diection of the acceleation vecto. The acceleation is the change in the velocity. It has to continuously change the velocity vecto back towads the cente of the cicle to keep the object moving in a cicle. Centipetal Foce Since we have an acceleation towad the cente of the cicle, thee must be a foce, The centipetal foce is just: F c = m v2 fom Newton s second law. The diection of the centipetal foce is the same and the centipetal acceleation i.e. inwad towad the cente of the cicle. Let s do poblems 6-15 and 6-20 as examples Un-Banke Cuves When a ca tavels aound an un-banked cuve, the foce holding the ca in the cuve is the static fiction between the ties and the oad. When we have olling without slipping it is static fiction that acts since thee is no motion between the bottom of the tie and the oad. N F f W Since N is just mg in this case, the maximum fiction foce is F s = µmg. The fiction is what povides the centipetal foce F c so: F c = m v2 = µmg µ = v2 g If the coefficient of fiction is not equal o lage than v2, the ca can not make the g cuve.
Banke Cuves When a ca goes aound a cuve on the highway, the cuve is banked. This gives the ca a component of the nomal foce fom the oad towad cente of the cicle. This component is the centipetal foce which makes the ca go aound the cicula cuve in the oad. F N F c W Fom the dawing, the centipetal foce is: F c = F N sin = m v2 Thee is no acceleation in the vetical so the weight is balanced by the vetical component of the nomal foce: F N cos = mg = W F N = mg/cos If we combine these two equations we have: tan = v2 g Let s do poblem 6-27 as an example
Fictitious Foce and Non-inetial Fames: Coiolis Foce In unifom cicula motion, thee may be constant speed and a constant magnitude of the acceleation but the vecto acceleation is changing because its diection is changing. Since the (vecto) velocity is changes, and object in unifom cicula motion is not in an inetial fame. An obseve in unifom cicula motion expeiences false foces. This is the so-called centifugal foce you feel when you go aound a cuve. You inetial wants you to go staight but you tavel with the ca aound the cuve. You feel a foce pushing you to the outside of the cuve. The Coiolis foce is the false foce on a huicane that cause it to otate counteclockwise in the nothen hemisphee Newtons Univesal Law of Gavitation Besides Newton s thee laws, Newton developed the law of gavity. Histoian tell us an apple did not fall on his head but it makes a nice stoy! Newton s law of gavity is: F gavity = G m 1m 2 2 The foce between two objects due to gavity is popotion to the poduct of the two masses divided by the distance between the objects squaed. The constant G = 6.67 x 10 11 N m 2 /kg 2, and is called the univesal gavitationalconstant. Notice by Newton s thid law that the foce on 1 due to 2 is the same magnitude but opposite diection fom the foce on 2 due to 1. When we ae nea the suface of the eath i.e., does not change much, we can think of the G m 1 pat of the equation as a constant assuming m 2 1 = m eath is the mass of the eath. Then: F gavity = m g Whee g = G m eath 2. This value g is ou old fiend 9.8 m/s 2. g = G m eath 2 g = 6.67 x 10 11 N m 2 /kg 2 5.98x1024 kg (6.38x10 6 m) 2 g = 9.8 m/s 2
We have come at this is a new way but it should not supise you that since F = m a, and the acceleation is -9.8 m/s 2, the foce on an object nea the suface of the eathis the object smasstime g. The foceon an object due to gavityiscalled weight. The unit is the Newton (N). Weight and mass ae not the same thing. In eveyday language we often intechange them but weight is a foce (vecto) and mass is a scala. An object fa fom anything in oute space has mass but it does not have weight. A point I find inteesting is that gavitation mass is exactly equal to inetial mass. The mass we use in Newton s law of gavity is the same as what we use in the Newton second law. Let s wok poblems 6-33 and 6-36 as examples Tides When the moon s gavity pulls on the eath, it distots the eath s shape to stetch out fom the sphee we nomally think of fo the eath. This distotion is about 1-2 m. Why ae thee two tides 12 hs apat? The moon s pull is weake on the opposite side fom the moon, nomal in the middle and stongest on the side neaest the moon. The eath gets elongated into a spheoidal shape. The bumps on the defomed eath aise the sea level. eath moon The sun also contibutes to tides but the sun is much futhe away so the effect is smalle. Sometimes the sun adds to the moon s tidal effect. Othe times it cancels out some of the moon s effect.
Satellites in Cicula Obits Fg satellite eath An eath satellite (m s ) in a cicula obit is in unifom cicula motion. The centipetal foce is supplied by gavity in the fom of Newton s law of gavity. F c = G m sm e 2 = m sv 2 whee is the adius of the obit, M e is the mass of the eath, m s is the mass of the satellite and G is the univesal gavitational constant. If we solve fo v: v = ( GM e ) 1/2 notice that the mass of the satellite canceled out of the equation. The obital speed depends only on the adius and planet s mass. Keple s Laws Befoe Newton, Johannes Keple using the obsevations of Tycho Bahe had woked out empiical elationships fo the motion (then) known planets. Undestanding the motion of the planets had been a poblem in astonomy since ancient times. Keple s law can be stated as: 1. Planets follow elliptical obits, with the sun at one focus of the ellipse. (Ellipses ae oval shaped closed cuves... a squashed cicle.) Note that will the obits ae ellipse, most of the planets, including eath have only slightly elliptical obits i.e., thei obits ae almost to cicles 2. As a planet moves in its obit, a line between the planet and the sun sweeps out equal aeas in equal times. 3. The peiod, T, of a planet is popotional to its mean (aveage) distance fom the sun to the 3 2 powe. As a fomula this can be stated as T 3 2
Keple s thid law is the most useful of Keple s thee laws. It follows diectly fom Newton s law of gavity. Gavity is what povides the centipetal foce to keep the planet in obit aound the sun: Since v = 2π T we have: F gavity = Centipetal Foce G m sm p v = m 2 2 p G m s = (2π/T)2 2 G m s = 4π2 2 T 2 T 2 = 4π2 3 Gm s T = 2π Gms 3 2 Which is Keple s 3 d law as an equality with the constant of popotionality. Let s wok poblem 6-43 and 6-49 as examples