1. Mechanics
Gravitations
MARLON FLORES SACEDON

2. The answer is Gravitations
Questions
Why doesn’t the moon fall to earth?
Why do the planets move across the sky?
Why doesn’t the earth fly off into space rather than remaining in orbit around the sun?
The answer is Gravitations

3. Gravitations What is Gravitations?
Gravity or gravitation is a natural phenomenon by which all things with energy are brought toward one another, including stars, planets, galaxies and even light and sub-atomic particles.

4. Gravitations Laws of motion of planets by Johann Kepler:
Law#1. The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)

5. Gravitations Laws of motion of planets by Johann Kepler:
Law#2. An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas).

6. Gravitations Laws of motion of planets by Johann Kepler:
Law#3. The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)
𝑻 𝟐 𝑹 𝟑

7. Gravitations Laws of motion of planets by Johann Kepler:
NOTE: The average distance value is given in astronomical units where 1 a.u. is equal to the distance from the earth to the sun x 1011 m. The orbital period is given in units of earth-years where 1 earth year is the time required for the earth to orbit the sun x 107 seconds. )

8. Gravitations There must be a force acted on the apple…
It was a force coming from earth which is a gravitational force.

9. Gravitations The Law of Universal Gravitations
The Law of Universal Gravitations by: Isaac Newton
The Law of Universal Gravitations
“Any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.” 𝑟
𝐹 𝑔 ∝ 1 𝑟 2
𝑚 𝐵
𝐹 𝑔 ∝ 𝑚 𝐴 𝑚 𝐵
𝐹 𝐴𝐵
𝐹 𝑔 =𝐺 𝑚 𝐴 𝑚 𝐵 𝑟 2
Where:
𝐹 𝐵𝐴
𝐹 𝑔 = gravitational force (N, dyne, poundal)
𝑚 𝐴
𝑚 𝐴 & 𝑚 𝐵 = interacting masses (kg, g, lb)
𝑟 = distance between masses (m, cm, ft)
𝐹 𝐵𝐴 − 𝐹 𝐴𝐵 =0
𝐺 = 6.67𝑥 10 −11 𝑁. 𝑚 2 𝑘𝑔 2 gravitational constant
𝐹 𝐵𝐴 = 𝐹 𝐴𝐵 = 𝐹 𝑔

10. Gravitations The Law of Universal Gravitations by: Isaac Newton
𝐹 𝑔 =𝐺 𝑚 𝐴 𝑚 𝐵 𝑟 2
𝐺 = 6.67𝑥 10 −11 𝑁. 𝑚 2 𝑘𝑔 2 gravitational constant
Cavendish balance
By: Henry Cavendish
Torsion Balance

11. Gravitations
Application Problem 1. Gravitational force of two interacting objects
Calculate the force of gravity on two interacting objects whose masses are 8kg and 15kg separated at distance of 2m. 𝑟
𝐹 𝑔 =𝐺 𝑚 𝐴 𝑚 𝐵 𝑟 2
𝐺 = 6.67𝑥 10 −11 𝑁. 𝑚 2 𝑘𝑔 2
𝑚 𝐵
𝐹 𝐴𝐵
𝐹 𝐵𝐴
𝐹 𝑔 =6.67𝑥 10 −11 𝑁. 𝑚 2 𝑘𝑔 2 ∙ 8𝑘𝑔 15𝑘𝑔 2𝑚 2 =2𝑥 10 −9 𝑁
𝑚 𝐴

12. Gravitations
Application Problem 2. Superposition of gravitational forces
Many stars belong to system of two or more stars held together by their mutual gravitational attraction. Figure below shows a three star system at an instant when the stars are at the vertices of a 45 𝑜 right triangle. Find the total gravitational fore exerted on the small star by the two large ones.
Calculate 𝑠 12 the distance between 𝑚 1 and 𝑚 2
𝑠 12
𝑠 12 = 𝑥 𝑥 =2.83𝑥 𝑚 +𝑦 +x 2
8.00𝑥 𝑘𝑔
45 𝑜
2.00𝑥 𝑚
Calculate the forces 𝐹 21 and 𝐹 31
𝐹 21 =6.67𝑥 10 −11 𝑁. 𝑚 2 𝑘𝑔 2 ∙ 8.00𝑥 𝑘𝑔 𝑥 𝑘𝑔 𝑥 𝑚 2
=6.67𝑥 𝑁
𝐹 21
𝐹 31 3
8.00𝑥 𝑘𝑔 1
1.00𝑥 𝑘𝑔
2.00𝑥 𝑚
𝐹 31 =6.67𝑥 10 −11 𝑁. 𝑚 2 𝑘𝑔 2 ∙ 8.00𝑥 𝑘𝑔 𝑥 𝑘𝑔 𝑥 𝑚 2
=1.33𝑥 𝑁

13. Gravitations
Application Problem 2. Superposition of gravitational forces
Many stars belong to system of two or more stars held together by their mutual gravitational attraction. Figure below shows a three star system at an instant when the stars are at the vertices of a 45 𝑜 right triangle. Find the total gravitational fore exerted on the small star by the two large ones.
𝑠 12
Σ 𝐹 𝑥 =1.33𝑥 𝑁+6.67𝑥 𝑁cos45=1.81𝑥 𝑁 +𝑦 +x 2
8.00𝑥 𝑘𝑔
45 𝑜
Σ 𝐹 𝑦 =6.67𝑥 𝑁sin45=4.72𝑥 𝑁
𝐹 𝑔 = (1.81𝑥 ) 2 + (4.72𝑥 ) 2 =1.87𝑥 𝑁
𝜃= 𝑇𝑎𝑛 − 𝑥 𝑥 = 𝑜
𝐹 21 =6.67𝑥 𝑁 3
8.00𝑥 𝑘𝑔 1
1.00𝑥 𝑘𝑔
𝐹 31 =1.33𝑥 𝑁

14. Gravitations
Solar System

15. Gravitations
Application Problem 2. Weight and Gravitational acceleration
𝐹 𝑔 =𝐺 𝑚 𝐴 𝑚 𝐵 𝑟 2
𝐹 𝑀𝑚
𝐹 𝐸𝑚
𝐹 𝑚𝑀
𝐹 𝐸𝑀
𝑭 𝒎𝑬
𝐹 𝑀𝐸

16. Gravitations
Application Problem 2. Weight and Gravitational acceleration
𝐹 𝑔 =𝐺 𝑚 𝐴 𝑚 𝐵 𝑟 2
𝐹 𝐸𝑚 =𝐺 𝑚 𝐸 𝑚 𝑚 𝑅 𝐸 2
Force of gravity exerted by earth to the man
𝑤 𝑚 =𝐺 𝑚 𝐸 𝑚 𝑚 𝑅 𝐸 2
𝐹 𝐸𝑚
Weight of the man
𝑅 𝐸
But weight is equal to mass time gravitational acceleration…
𝑤 𝑚 = 𝑚 𝑚 𝑔
𝑭 𝒎𝑬
𝐹 𝐸𝑚 =𝑤 𝑚
𝑔 𝑀 =𝐺 𝑚 𝑀 𝑅 𝑀 2
𝑚 𝑚 𝑔 𝐸 =𝐺 𝑚 𝐸 𝑚 𝑚 𝑅 𝐸 2
Gravitational acceleration at the surface of the moon
𝑔 𝐸 =𝐺 𝑚 𝐸 𝑅 𝐸 2
Gravitational acceleration at the surface of the earth

17. GravitationAL POTENTIAL ENERGY

18. MOTION OF SATELLITES

19. Gravitations Problem Exercise/ Seatwork/ Assignment
1. The moon has a mass of 7.32 𝑥 𝑘𝑔 and a radius of 1,609.4 𝑘𝑚. Calculate the value of "g" at the surface of the moon.
2. At what distance from the earth will a spacecraft on the way to the moon experience zero net force because the earth and moon pull with equal and opposite forces?
3. Calculate the force of gravity on a spacecraft 12,800 𝑘𝑚 above the earth’s surface if its mass is 850 𝑘𝑔.
4. Calculate the centripetal acceleration of the earth in its orbit around the sun and the net force exerted on the earth. What exerts this force on the earth? Assume the earth’s orbit is a circle of radius 1.49 𝑥 𝑚. P A B
50𝑐𝑚
30𝑐𝑚
35 𝑜
5. Two uniform spheres, each of mass 𝑘𝑔, are fixed at point 𝐴 and 𝐵. Find the magnitude and direction of gravitational attraction force on a uniform sphere with mass 𝑘𝑔 at point 𝑃.

20. PROBLEMS

21. PROBLEMS

22. PROBLEMS

23. PROBLEMS

24. PROBLEMS

25. PROBLEMS

26. PROBLEMS

27. PROBLEMS

28. ANSWERS TO ODD NUMBERS

29. eNd