Extra notes for circular motion: Circular motion : v keeps changing, maybe both speed and

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1 Exta notes fo cicula motion: Cicula motion : v keeps changing, maybe both speed and diection ae changing. At least v diection is changing. Hence a 0. Acceleation NEEDED to stay on cicula obit: a cp v /, pointing to the cente along adius diection. Total foce NEEDED in the adius diection to stay and maintain cicula motion: F needed ma cp mv /, pointing to cente 1. When total foce in diection pointing to cente equals to mv /, it has what it needs to stay in obit. ΣF mv /, it stays in cicula obit. Neithe flies away, no falls into the cente. This is how you solve elated foces. Velocities o adius, etc. 1. Find ALL REAL Foces and foce components,. ADD ALL in adius diection ΣF, towad cente. 3. Make ΣF At that position equals to mv / at that position 1 4. Solve Equation ΣF mv/

2 Chapte 1 Gavity Whee does mg come fom? Why is g9.81m/s on eath suface? Sample poblems: What speed is needed to launch a object to fly aound eath (close to eath suface)? Whee ae the synchonous satellite (fo TV) located? How to find the peiod of a planet obiting aound How to find the peiod of a planet obiting aound the Sun?

3 1-1 Newton s Law of Univesal Gavitation The foce acceleating an apple downwad is the same foce that keeps the Moon in its obit. Hence, Univesal Gavitation. 3

4 1-1 Newton s Law of Univesal Gavitation The gavitational foce is always attactive, and points along the line connecting the two masses centes: is the distance between mass centes of m 1 and m F g G M m M 1 1 G m The two foces shown ae an action-eaction pai. 4

5 1-1 Newton s Law of Univesal Gavitation G6.67 x Nm /kg, the univesal gavitational constant G is a vey small numbe; this means that the foce of gavity is negligible unless thee is a vey lage mass involved (such as the Eath). M 1 m M 1 F g G G m When m 1 o m inceases F g inceases. When inceases, F g deceases. F g is popotional to 1/ Attention: ti 1 ove squae! Not 1 ove. 5 When is doubled, F g will educe to 1/4 of initial F g.

6 1- Gavitational Attaction of Spheical Bodies Gavitational foce between a point mass and a sphee: the foce is the same as if all the mass of the sphee wee concentated at its cente. 6

7 1- Gavitational Attaction between Eath and an object on it The cente of the Eath is one Eath adius away fom object m, so this is the distance we use: Theefoe, g G mm Eath F g G R M Eath Eath R Eath M Eath 5.97 *10 R Eath 6.37 * m kg Eath g Eath 9.81 m / s 9.81 N / When m is on othe planets M p, R p ae diffeent. g planet will be diffeent fom 9.8 m/s 7 kg mg Eath

8 Example 1: Cicula obit aound eath To let an object fly aound the eath close to eath suface, you need to launch it with what velocity? In the adius diection, Foce it has what s needed F net along adius mg m v / v g, so v g g. so, v g. Plug in numbes: Close to eath suface. g9.81 m/s Eath adius 6370km 6.37 x 10 6 m V 7.9 x 10 3 (m/s) ~ 5 mile/s Time to cicle the wold : T π / v π 6.4 x 10 6 / 7.9 x x 10 3 (s) 85 (min) 8

9 1- Gavitational Attaction of Spheical Bodies The acceleation of gavity deceases with altitude: 9

10 1- Gavitational Attaction of Spheical Bodies Once the altitude becomes compaable to the adius of the Eath, the decease in the acceleation of gavity is much lage: F g is popotional to 1/ 10

11 Example, What is the distance h between synchonous satellite (fo TV) and the eath suface? Synchonous satellite otates aound eath cente at the same peiod as eath does, 4 hous. To keep cicula obit, In the adius diection, The net Foce the satellite has what s needed R + 4h Eath h s G m M satellite Eath π / v T s M G Eath v v π / T GM Eath T m satellite v satellite M M (π) G Eath T 3 4π 4.E7 m, h3.6e7m 11 Wow, all synchonous satellites ae 36000km above eath suface.

12 Example 3, How to find the peiod of a planet obiting aound the Sun? Assuming the obit is a cicle. Again, to keep cicula obit, in the adius diection, The net Foce the planet has what s needed G m M Planet SUN SuntoPlaen t m Planet v SuntoPlaen planet t M G SUN v SuntoPlaen t v planet v π / T T π / You don t need to memoize any esults above. Mass of the sun kilogams You only need to lean and solve ΣF F g mv / Use the coect, coect mass and coect v. 1

13 Summay of Chapte 1 Foce of gavity between two point masses: G is the univesal gavitational constant: In calculating gavitational foces, spheically symmetic bodies can be eplaced by point masses. 13

14 Summay of Chapte 1 Acceleation of gavity on eath : Combine gavity foce with cicula motion fo obits. Gavitational foce GmM/ mv / What speed is needed to launch a object to fly aound eath (close to eath suface)? Know mg and, find v. Whee ae the synchonous satellite (fo TV) located? Know T, Eath mass, G, find ; (know vπ/t) How to find the peiod of a planet obiting aound the Sun? Know Sun mass, G, ; find v and T (know Tπ/v) 14

Gravitation. AP/Honors Physics 1 Mr. Velazquez

Gravitation. AP/Honors Physics 1 Mr. Velazquez Gavitation AP/Honos Physics 1 M. Velazquez Newton s Law of Gavitation Newton was the fist to make the connection between objects falling on Eath and the motion of the planets To illustate this connection