Uniform circular motion & orbits

uniform circular motion

• the object moves in a circle at constant speed v

• the magnitude of acceleration is:

  • a = v² / r

• the objects velocity and acceleration are NOT constant because the direction is always changing

a = v² / r

what is the magnitude of the acceleration of uniform circular motion?

centripetal acceleration

what kind of acceleration is present while an object is in uniform circular motion?

towards the center of the circle

in what direction is centripetal acceleration?

centripetal force

• a net force pointing towards the center of the circle to keep the object moving in a circle 

• “pulling” the object towards the center of the circle

towards the center of the circle 

in what direction is centripetal force?

Fnet = ma = mv² / r

what is the equation for the magnitude of centripetal force?

Newton’s second law tells us that the Fnet on the object is the mass times its acceleration

remember: Newton’s second law tells us that the Fnet on the object is the mass times its acceleration

Newton’s law of universal gravitation

• there is a gravitational force between any two masses

• mass 1 being pulled towards mass 2 & mass 2 being pulled towards mass 1

  • two masses are attracted to each other

  • the forces on each mass are equal & opposite

  • the two forces form an action/reaction pair (newton’s 3rd law)

F1on2 = F2on1 = Gm1m2 / r² 

what is the equation for the magnitude of the gravitational force btw two masses?

the universal gravitational constant

what is G in the equation to find the magnitude of the gravitational force?

G = 6.673 × 10^-11 Nm²/kg²

what is the value of the universal gravitational constant (G)?

different planets have different masses & radii, so there is a different acceleration due to gravity (g) on the surface of each planet

remember: different planets have different masses & radii, so there is a different acceleration due to gravity (g) on the surface of each planet 

gplanet = GMplanet / R²planet

what is the equation to find the accel. due to gravity on ANY planet?

GMplanet / R²planet = gplanet in F = mgplanet

remember: GMplanet / R²planet = gplanet in F = mgplanet

if you throw a projectile horiz. from a given height above the surface of the earth, it will move horiz. @ constant speed and it will fall vertically w/ a constant accel.

remember: if you throw a projectile horiz. from a given height above the surface of the earth, it will move horiz. @ constant speed and it will fall vertically w/ a constant accel.

if you throw a projectile really fast, it will go so far that the ground will be curving away from it 

remember: if you throw a projectile really fast, it will go so far that the ground will be curving away from it 

Orbits

• if you throw a projectile @ just the right speed it will fall all the way around the planet and keep circling around the planet as it falls (bc the curvature of its trajectory matches the planet

• a force pulling the projectile towards the center of the circle

  • this force is gravity

    • the force of gravity = the centripetal force

    • so Mag. of the force of gravity = Mag. of the centripetal force

    • GMm / r² = mv² / r

Gravity = Centripetal force (GMm / r² = mv² / r) 

remember: Gravity = Centripetal force (GMm / r² = mv² / r) 

v = √GM/r

how to find the speed at which the satellite/projectile is moving in order to move in that orbit (what equation)?

Period (T)

• time for one full rotation around the circle

• T = 2πr / v

T = 2πr / v 

what is the equation to find the period (T)? 

2πr

circumference of a circle 

the satellites/ projectiles velocity vector is tangent to the circle, its accel is towards the center of the circle, & the its being pulled towards the earth by gravity

remember: the satellites/ projectiles velocity vector is tangent to the circle, its accel is towards the center of the circle, & the its being pulled towards the earth by gravity

because its being pulled towards the earth by gravity

why does a satellite move in a circle?

the satellite must have the speed √GM/r to maintain a circular orbit of radius r

remember: the satellite must have the speed √GM/r to maintain a circular orbit of radius r