AP Physics C Mechanics Memorize

The mathematical equations for position of a SMH

y = Asin(wt + O) where w = 2pi*f

The mathematical equations for velocity of a SMH

y = Awcos(wt + O) where w = 2pi*f

The mathematical equations for acceleration of a SMH

y = -Aw²sin(wt + O) where w = 2pi*f

The differential equation () has the solution of y = Awcos(wt + O) where w = 2pi*f

d²y/dt² = -w²y

The derivation process for the differential equation and thus the position function is…

F = ma

-kx = m(d²x/dt²)

-k/m * x = d²x/dt²

Then solve for x(t)

The differential equation for a pendulum is

θ(t) = θmax sin(wt + O) where w = sqrt(g/L)

Work for rotiations

W = integral from θ1 to θ2 of torque times dθ

Power for Rotations

P = torque times angular velocity

How to convert linear momentum into angular momentum

multiple the effective angular momentum by radius from axis of rotation (rmv)

Rotational KInetic Energy for a pendulum is…

Neglected, so dont include in when doing energy equations.

The rotational energy is neglected because pendulum’s diameter is small and angular displacement is very small.

K effective for a block connected by 2 parallel springs

k1 + k2

K effective for a block connected by 2 springs on different sides of the block (180 degrees from each other)

k1 + k2

K effective for a block connected by 2 springs in series

(k1 k2) / (k1 + k2)

Kepler’s 3 Laws

1. Every Planet moves in an elliptical orbit, with the Sun at one focus

2. As a planet moves in orbit, a line drawn from the Sun to the planet sweeps out equal areas in equal time intervals —> Conservation of Angular Momentum

3. It T is the period, the time required to make one revolution, and a is the length of the semimajor axis of a plane’ts orbit, then the ratio T²/a³ is the same for all planets orbiting the same star.

If an orbit is circular

Then the planet’s orbit speed must be constant

Expanding on Kepler’s Third Law for circular motion: T²/R³ (Where R is the distance from the center of the planet to the other object) =

4(pi)²/(GM) where M is the Mass of the planet creating the gravity

For Gravity only the mass () us exerts a gravitational force

Underneath. This property is used alongside volume density to find the effective Mass or the Mass within.

The Period of a planet’s orbit is…

T = 2(pi)R / v where v is the planet’s orbit speed and R is the distance from the center of the planet to the object.

Escape speed for any planet is…

sqrt(2GM/r) where r is the radius of the Earth and M is the mass of the planet

To find the escape velocity for any planet…

½ mv² - GMm / r = 0 where r is the radius of the planet and M is the mass of the planet

The Expression for the total energy of a satellite in a circular orbit of radius R

E = -GMm / (2R) and this can be generalized for elliptical orbits as E = -GMm / (2a) where a is the length of the semimajor axis. The length of the semimajor axis is basically the HALF OF THE TOTAL LENGTH ACROSS of an ellipse OR from the center of an ellipse to one of its end.

A Barycenter is…

When one body orbits another, both bodies orbit around their center of mass, a point called the barycenter

For a system of two planets…

The centripetal force on each body is provided by the gravitational pull of the other body.

The more general version of Kepler’s Third Law for two bodies orbiting a barycenter (usually the Sun) is…

T²/(R1 + R2)³ = (4pi²) / (G(M1 + M2))

The force of gravity equation for inside a planet is…

GMmr / R³ where r is the distance from the core to the current location and R is the total radius of the planet

Integral formand summartion form for I

integral of r² dm where dm = M/L dx

Summation of all mr²

Integral and summation form of Xcm

1/M time integral of x dm where dm = M/L dx

and M = Integral of M/L dx