AP PHYSICS C: MECHANICS



AP PHYSICS C: MECHANICS   

|Linear Forms |Rotational Forms |

---------------------Kinematics---------------------

|[pic] = Position ([pic] - initial ; [pic] - final) |[pic] = Position ([pic] - initial ; [pic] - final) - radians |

|[pic] = Displacement |[pic] = Angular displacement ([pic] = arc length) |

|[pic] = Velocity ([pic] - initial ; [pic] - final) |[pic] = Angular velocity ([pic] - initial ; [pic] - final) |

|[pic] = Acceleration |[pic] = Angular acceleration |

|[pic] = Velocity of a with respect to b |[pic] = Radius |

| |[pic] = frequency (hertz --> Hz: amt. per sec.) |

| |[pic] = period (seconds) |

|Defnintion of [pic]: |Definition of [pic]: |

| [pic]  | [pic]  |

|Definitions of [pic] and [pic]: |Definitions of [pic] and [pic]: |

| [pic]  | [pic]  |

|Derivation (slope of tangent) to get [pic]/[pic]: |Derivation (slope of tangent) to get [pic]/[pic]: |

|  [pic]  | [pic]  |

|Integration (area under graph) to get [pic]/[pic]: |Integration (area under graph) to get [pic]/[pic]: |

|  [pic]  | [pic]  |

|Kinematic Equations 1&2: |Kinematic Equations 1&2:  (by analogy to linear equivalents) |

|[pic] |Constant [pic] graph: |

|[pic] | [pic]  |

| |Constant [pic] graph: |

|Equation of graph: | [pic]   |

|(constant v) | |

|[pic] |Kinematic Equation 3:  (by analogy to linear equivalent) |

|or alternatively: | [pic]  |

|  [pic]  | |

|Equation of graph: |Kinematic Equation 4: |

|(constant a) |Combining 2 and 3 to eliminate [pic] yields: |

|[pic] |combining yields:  [pic]  |

|or alternatively |[pic] |

| [pic]  |Connecting linear values to rotational ones:  [pic] and [pic] are |

| |tangent to circle. |

|Kinematic Equation 3: | [pic]  |

|[pic] |[pic] and [pic] are vectors along axis of rotation with direction |

|Sum of areas: |determined by right hand rule |

|[pic]  |[pic] |

|Area of q: |Frequency & Period: |

| | [pic]  |

|Area of r: | [pic]  |

|[pic] | |

|Therefore:  | |

|[pic] | |

| [pic]   | |

| | |

|Kinematic Equation 4: | |

|Combining 2 and 3 to eliminate [pic] yields: | |

| [pic]  | |

|[pic] | |

|Reference Frame Addition:  [pic]  | |

|[pic] reads as "velocity of a with respect to b."  Treat | |

|x and y directions separately. | |

---------------------Forces/Torques---------------------

|[pic]= sum of the forces (Newtons) |[pic] = moment of inertia (rotational analog of [pic] for |

|[pic] = weight force (Newtons) |torques) Values of [pic]: |

|[pic] = normal force (Newtons) |Point mass moving in circle [pic][pic] |

|[pic] = friction force (Newtons) |Thin Hoop: [pic] |

|[pic] = coefficient of static friction |Rod length [pic] about its center [pic] |

|[pic] = coefficient of kinetic friction |Solid disk/cylinder radius [pic]: [pic] |

|[pic] = applied force (Newtons) |Solid sphere radius [pic]: [pic]  |

|[pic] = tension force (Newtons) |Hollow sphere [pic]  |

|[pic] = elastic force (Newtons) |[pic]= torque (analog of [pic]) |

|[pic]= centripetal acceleration ([pic]) |[pic] = angular momentum (analog of [pic]) |

|[pic]= acceleration due to gravity ([pic]) |[pic] = angular equivalent of impulse.  Has no letter to represent |

|[pic] = momentum ([pic]) |it. (analog of [pic]) |

|[pic]= impulse ([pic]) | |

|Forces: |[pic](statics) |

|[pic]  (statics) | [pic] (acceleration) |

| [pic]  (acceleration) |Torque is a vector along the axis of rotation with direction |

|Treat x and y force components separately and then combine components |determined with right hand rule. |

|with Pythagorean theorem: |Parallel Axis Theorem: [pic] , where [pic] is the old moment of |

|[pic] |inertia, and [pic] is the distance between the old parallel axis of |

|[pic] |rotation and the new axis. |

|[pic] |[pic] |

|[pic] |Momentum & Impulse (by analogy): |

|Momentum & Impulse: |  |

|[pic] [pic] |[pic] [pic]  |

|[pic]Also... |Also... |

|[pic] | [pic]  |

|[pic]  |[pic] |

|(i.e. - Impluse is the area under a [pic]graph) | [pic]  |

| |And... |

| | [pic]  |

| |[pic] |

| |Calculating moments of inertia: |

| | [pic]  |

---------------------Energy--------------------- 

|[pic] = Potential Energy (Joules) |[pic] = Rotational Kinetic Energy (Joules) |

|[pic] = Kinetic Energy (Joules) | |

|[pic] = Elastic Potential (Joules) | |

|[pic] = Work (Joules) | |

|[pic] = Power (Watts) | |

|[pic] indicates dot product | |

|[pic]  |[pic] |

|[pic] [pic]  [pic]   | |

| [pic]  | |

| [pic]  | |

|Also, since [pic] | |

| [pic]  | |

| [pic]  (work done by field) | |

---------------------Center of Mass---------------------

(i.e. - sum of mass times position of all particles divided by sum of mass of all particles)

---------------------Gravity---------------------

• [pic] = semimajor axis length

• [pic] = apogee radius

• [pic] = perigee radius

• [pic] = universal gravitational constant [pic]

• [pic] = mass of larger object

• [pic] = mass of smaller object

• [pic] = distance between two objects OR radius of planet

• [pic] = distance to center of planet

• [pic] = velocity of orbiting object

• [pic] = escape velocity

[pic]

 [pic] = constant for all orbits around same object

Fg adjusted for planetary motion:

 [pic] 

gravitational acceleration as a function of planet radius:

 [pic] 

gravitational acceleration as an object moves inside a planet:

 [pic] 

adjusted Eg for any location above planet's surface, r measured from center of planet:

 [pic] 

orbital velocity as a function of r:

  

energy is conserved throughout orbit (value is negative because it is bound):

 [pic] 

  

escape velocity (projectile launched perpendicular to earth's surface):

[pic], or in other words

[pic], and since [pic], then we can say

[pic], and thus,

 [pic] 

---------------------Simple Harmonic Motion (SHM)---------------------

 [pic] 

 [pic] 

 [pic] 

([pic] = angular frequency [pic])

 [pic]  and thus  [pic] 

By substitution, [pic]

Solving for angular frequency gives: [pic] 

Substituting into [pic] gives [pic]  

Period of a SHM pendulum (15 degrees or less):

[pic] = length of string  

 [pic] 

---------------------Velocity Dependent Friction---------------------

terminal velocity = [pic] 

[pic] = some constant that will be given to you or you will have to solve for

differential equation madness! 

Example: [pic], where [pic] is a constant and [pic] is the velocity of the object going through air.  Let's say it's falling from rest, and has mass [pic].  Find terminal velocity.

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

let [pic], so thus [pic], so [pic]

[pic]

MEMORIZE THIS: [pic]

[pic]

Constants of integration can really be combined into one constant of integration.

[pic]

[pic]

[pic]

[pic]

[pic]

note that when [pic], [pic], and thus [pic], so [pic]

[pic]

[pic]

[pic]

 [pic] 

note here that the constant is [pic]

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