Force of Gravity

Mass

Mass is a measure of an object's resistance to acceleration from a force. Low mass objects will accelerate more from the same force while high mass objects will accerelate less. A force that can move a bicycle will barely budge a car.

Gravity

Mass also determines the strength of the force of gravity. All massive objects are attracted towards each other, but we mostly notice the attraction towards the earth because it is so large and so close.

Another word for the force of gravity is weight. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass.

Fg = mg

Fg = the force of gravity, weight [N, Newtons, nonmetric: pounds] vector
m = mass [kg]
g = acceleration of gravity on the surface of the earth = 9.8 [m/s²] vector

The Force of gravity

The force of gravity depends on the mass of the object, but on the surface of the earth acceleration from gravity is the same for all objects.

So why do feathers fall slower then bricks?



Technically the acceleration of 9.8 m/s² on the surface of the earth is just an approxomation. Gravity actually changes depending on where you are.

What is the force of gravity in Los Angeles?



Why would the acceleration of gravity seem lower near the equator?





Example: The hardcover version of the book Seveneves by Neal Stephenson has a mass of 0.95kg. What is the force of gravity felt by the book on earth? What about on the moon?

solution

weight on earth:

$$F_{g}=mg$$ $$F_{g}=0.95(9.8)$$ $$F_{g}=9.31N$$

weight on moon:

$$F_{g}=ma$$ $$F_{g}=0.95(1.6)$$ $$F_{g}=1.52N$$


Example: The book Uprooted by Naomi Novik is resting on a table. Amazon says the shipping weight is 1.2 pounds.

  • What is the mass in kilograms of the book on earth? On mars?
  • What is the weight of the book in Newtons on earth (a = 9.8 m/s²)? On mars (a=3.711 m/s²)?

    solution

    mass is the constant everywhere

    $$ 1.2lbs \frac{1kg}{2.2lbs} = 0. \overline{54}kg $$ $$ m = 0. \overline{54} kg $$

    weight on Earth

    $$F_{g}=mg$$ $$F_{g}=(0.\overline{54})(9.8)$$ $$F_{g}=5.35N$$

    weight on Mars

    $$F_{g}=ma$$ $$F_{g}=(0.\overline{54})(3.711)$$ $$F_{g}=2.02N$$






    Weightless

    Earth's gravity does extend into space, but it decreases with distance. It is about ~90% for astronauts in orbit around earth, but they don't notice any gravity because they are in a freefall.

    Freefall

    Freefall means that you are just letting gravity accelerate you without any opposing forces. To keep from falling we are carefull to always counter the force of gravity. This can be done with a parachute or a jetpack! Mostly we counter gravity with a normal force from the ground. mass Normal Force Force of gravity

    The Normal Force

    Normal means perpendicular. A normal force is perpendicular to the surface the object is in contact with. Typically a normal force will balance the force of gravity to keep an object from accelerating up or down, but not always.





    Example: You are accelerating up in an elevator at 2m/s². If your mass is 100kg what is the normal force you feel from the elevator?

    solution $$F_{g}=mg$$ $$F_{g}=(100)(-9.8)$$ $$F_{g}=-980N$$
    $$\sum F=ma$$ $$F_{g}+F_{N}=ma$$ $$F_{N}=ma-F_{g}$$ $$F_{N}=(100)(2)-(-980)$$ $$F_{N}=1180N$$






    Example: A 20kg box is at rest on a horizontal sidewalk. Find the force of gravity and the normal force on the box.

    solution 20kg FN = 196N Fg = 196N $$F_{g}=mg$$ $$F_{g}=(20)(-9.8)$$ $$F_{g}=-196N$$

    In the simple case of a flat horizontal surface and no vertical acceleration the force of gravity will be equal and opposite to the normal force, but not directly because of Newton's 3rd law.

    $$\sum F=ma$$ $$F_{g}+F_{N}=ma$$ $$F_{N}=ma-F_{g}$$ $$F_{N}=(20)(0)-(-196)$$ $$F_{N}=196N$$




    20kg FN Fg

    Example: A 20kg box is at rest on a steep sidewalk. The sidewalk is at an angle 20 degrees from horizontal. Find the force of gravity and the normal force on the box. What is the acceleration of the box? (assume no friction)

    solution $$F_{g}=mg$$ $$F_{g}=(20)(-9.8)$$ $$F_{g}=196N$$

    Separate the gravity vector into components parallel and perpendicular to the ground

    perpendicular to ground

    $$F_{g\perp}=F_{g}\cos(20)$$ $$F_{g\perp}=(196)\cos(20)$$ $$F_{g\perp}=184N$$
    $$\sum F_{\perp}=ma$$ $$F_{g\perp}+F_{N}=ma$$ $$F_{N}=ma-F_{g\perp}$$ $$F_{N}=(20)(0)-(-184)$$ $$F_{N}=184N$$

    parallel to ground

    $$F_{g\parallel}=F_{g}\sin(20)$$ $$F_{g\parallel}=(196)\sin(20)$$ $$F_{g\parallel}=67N$$
    $$\sum F_{\parallel}=ma$$ $$F_{g\parallel}=ma$$ $$(67)=(20)a$$ $$3.35\small\frac{m}{s^{2}}=\normalsize a$$





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