Universal Gravitation Gravitation Test Review

What is the equation for the Force of Gravity/weight acting on an object?

F_g = m_og

m_o- mass of object that gravity is acting on

Newton’s Universal Law of Gravitation

F_g = Gm₁m₂/r²

Vector Form: F_{g, vectorized} = Gm₁m₂/r² * r₁₂ where r₁₂ is the unit vector

another way of defining the force of gravity

Shows the force of gravitational attraction between any two masses

What is G?→ The universal gravitational constant

• G = 6.67 × 10^-11 Nm²/kg²

r is the distance between the centers of mass of the two objects

This equation gives you the magnitude of the force of gravitational attraction between two objects

Direction is always towards the other object

How to solve for the acceleration due to gravity?

GM/r²

What is the acceleration due to gravity for earth?

g_earth = Gm_earth/(r_earth+altitude)²

need to take into account the altitude!

as long as the radius of the earth is much much greater than the altitude, the acceleration due to gravvity is constant and thus the gravitational potential energy is constant as well U_g = mgh

When viewed from a frame of reference which is not on the surface of the planet, what happens to the acceleration due to gravity?

It isn’t constant and we need a different equation for GPE

Universal gravitational potential energy would be U_g = -Gm₁m₂/r

Key features:

• The zero line is at infinity so (r = infinity)

• The universal gravitational potential energy is always less than or equal to 0

• It requires two objects

Binding energy

The minimum amount of work necessary to completely remove an object from a planet if the object is resting on the surface of the planet

Have to assume the object is moved infintely far away and has zero velocity when it gets there

ΔE_system = ΣT = ΔME + ΔE_internal = W_{force applied}

where ΣT is the total energy transferred in and out of the system

Minimum amount of work:

W_{force applied} = ME_f-ME_i + 0 = -ME_i (doesn’t have any mechanical energy at end and no internal energy)

Therefore, W_f,a = GM_oM_p/R_p

**If anything less than this amount of energy is given to the object, object will return back to surface

Escape Velocity/Speed

The minimum velocity to launch an object off the Earth and have it never return

We assume that there is no atmosphere and no Earth rotation:

• Therefore no external forces acting on system so mechanical energy conserved

V_escape = √(2GM_E)/R_E

Orbital energy

Total mechanical energy of an object in circular motion

ME_tot = -Gm_om_p/(2r)

Kepler’s First Law

Law of orbits:

All orbits are elliptical

Check diagram for more details

Kepler’s Second Law

Law of areas:

• States that a line between the sun and an orbitting planet sweeps out an equal area in an equal time interval

• Means that the closer an object is to what it is orbitting, the faster the orbital object’s tangential speed will be (area depends on distance from center and velocity, meaning if the distance is lower, velocity must be higher to compensate)

Kepler’s Third Law

The Law of Periods (shows the relationship between the period of the object and its radius)

• Must recognize that the only force acting on the orbital object is the force of gravity and that force of gravity acts inward towards the center of the circle (circular orbit)