Mr. Rogers AP Physics C Study Guide: AP Physics C Mechanics 1st Semester Exam

 1st Quarter Topics > Kinematics(1) Vector Addition(2) Projectile Motion(2) Newton's Laws(3)
 2nd Quarter Topics > Newton's Laws(4) Friction(5) Mech Energy(7) Momentum
 Online Quiz - 1st Semester Exam: This quiz will is similar to your semester exam.                Q21A2-4CB9 Additional practice: work problems in both the Barron's and Princeton review study books (see syllabus). The information below is slanted specifically for the First Semester exam and has generally not been covered in other venues, but is also not all inclusive. Consult the study guides from previous units for the complete picture.

Mathematical models

Springs in Parallel   Springs in Series
Ktotal = K1 + K2 + K3+ ... + Kn   1 / Ktotal = 1 / K1 + 1 / K2 + 1 / K3+ ... + 1 / Kn

Universal Gravity Equation   Pendulum Oscillation
 F = [G(m2m1)] / r2 G = universal gravity constant r = distance between center of mass of  m2 & m1 The attraction forces acting on the 2 masses form an action reaction pair..
f = 1/(2p) (g / L)^0.5

T = 1 / f

(f = frequency, T = period)

Gravity Field Strength (Acceleration due to Gravity)   Spring and Mass System Oscillation
 g = [G(m2)] / r2 Plugging  in the radius and mass of Earth yields g = 9.8 m/s2
f = 1/(2p) (k / m)^0.5

T = 1 / f

Gravitational Potential Energy   Angular or Rotational Kinematics
 U = - m1[G(m2) / r] For 2 masses at indeterminate distance. usually  m1 represents an object and m2 a planet. Note U = 0 at infinity. The right side is negative since U must decrease as the object falls toward the planet

 d = rq q is the angular displacement in radians. There are 2p radians in a circle or in one revolution. v = rw w is angular velocity. a = ra a is angular acceleration w = at + wo q = 1/2 at2 + wot Kinematics equations for rotation are identical to  linear motion except angular parameters are substituted for linear ones.
Inertia   Momentum
 Linear m Rotational  (moment of inertia for a point mass rotating at a distance r from a center of rotation) I = mr2

 Linear P = mv Rotational L = Iw
 Key Principles
 Solving rotational kinematics problems is identical to solving linear kinematics problems except that the linear parameters of a, v, and d are replace with their rotational counterparts. Center of mass can be considered a balance point. For a seesaw, the center of mass is directly above the fulcrum when the seesaw is balanced.
 Example Problems
Finding g for an unknown planet

If an unknown planet has 1/2 the radius and twice the mass, what is the value of g on the planet.

g unknown  (2 / (1/2)2 )(g Earth)

=   8 (g Earth)

Explanation: The above comes directly from the universal gravity equation. Gravity force is directly proportional to mass and inversely proportional to the square of the distance between the centers of mass of the objects

Angular or rotational motion kinematics

A fan is switched on. It has an angular acceleration of 10 rad/s2. After 10 seconds, what is the fan's angular velocity and displacement?

w = at + wo

= (10 rad/s2)(10 s) + 0

q = 1/2 at2 + wot

= 1/2 (10 rad/s2)(10 s)2 + 0

Elliptical Orbits

For an elliptical orbit of a mass =m around a planet, the velocity at the closest point = vc and the closest distance = Rc with the furthest distance = Rf. What is the kinetic energy at the furthest distance?

First find the velocity vf

vf = (Rc/ Rf) vc

K = 1/2 m v 2

= 1/2 m [(Rc/ Rf) vc] 2

Pendulums

A pendulum with a mass of m and length of L swings with a frequency of 3 Hz. What will the frequency be for a pendulum with a mass of 3m and a length of 1/4 L?

f = 1/(2p) (g / L)^0.5

= 1/(2p) [g / (1/4 L)]^0.5

= 1/(2p) [4 g / L]^0.5

= (2) 1/(2p) [g / L]^0.5

but 1/(2p) [g / L]^0.5 = original f, therefore

f  = 2 (3 Hz)

= 6 Hz

Frequency of a Mass and Spring System

You have two linear springs with spring constants of k and 3k. You also have 2 masses m and 2m. What is the ratio of the highest to lowest resonant frequencies possible using these components.

For parallel springs:  Ktotal = (3k + k)

= 4k

for series springs:  1 / Ktotal = 1 / (3k) + 1 / k

Ktotal = 3 / 4 k

highest freq = 1/(2p) [4k/ m]^0.5

lowest freq  = 1/(2p) [(3 / 4) k / (4m)]^0.5

 (ratio high to low) = 1/(2p) [4k/ m]^0.5 1/(2p) [(3 / 4) k / (4m)]^0.5

 (ratio high to low) = [4]^0.5 [3 / 16]^0.5

 (ratio high to low) = 8 (3)^0.5

Rotational Momentum

A magical massless  ice skater holds a dumbbell with a mass of m in each hand as she spins around with a rotational velocity of  wo. The distance between the center of mass of the 2 dumbbells is 2R. She pulls her arms in so that the distance is R. Find her new rotational velocity.

from conservation of momentum:

Lbefore = Lafter

Ibefore  wo= Iafter w

2 [m (R)2] wo = 2 [m (1/2 R)2] w

wo = 1/4 w

w = 4 wo

 Vocabulary
 Period Angular or Rotational Displacement Frequency Angular or Rotational  Velocity Center of Mass Angular or Rotational Acceleration Linear Inertia Angular or Rotational Inertia (Moment of Inertia) Angular or Rotational Momentum
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