Chapter 7-9 Flashcards
Center of Mass (CM)
the average position of mass for a system of particles
Equation on equation sheet
Conservation of Linear Momentum
if the sum of the external forces acting on a system of particles is zero, then the linear momentum of the system must remain constant
-if sum(Fi) = 0 then p is constant since F = dp/dt = 0
Torque
or moment, is the rotational counterpart to force
tau = r cross F
Conservation of Angular Momentum
if the sum of the external torques acting on a system of particles is zero, then the angular momentum of the system must remain constant
-if sum(taui) = 0 then L is constant since tau = dL/dt = 0
Laboratory Coordinates
reference frame in which the origin is at rest in the laboratory
Center of Mass (CM) Coordinates
reference frame in which the origin moves with the center of mass
Reduced Mass
mu = m1 x m2 / (m1 + m2)
Coefficient of Restitution
epsilon = abs(v2’ - v1’) / abs(v2 - v1) = v’ / v
v’ / v = final relative speed / initial relative speed
Elastic Collision
coefficient of restitution (epsilon) is equal to 1, so Q = 0 and total Kinetic Energy (T) is constant
Inelastic Collision
coefficient of restitution (epsilon) is greater than or equal to 0 and less than 1, so Q > 1 and Kinetic Energy (T) is not constant
Rigid Body
non-deformable object (the distance between any two points is constant)
Center of Mass for a Rigid Body
rcm = int(r dm) / int(dm)
for a composite object:
rcm = sum(mi (rcm)i) / m
if a uniform body has a plane of symmetry (a plane that divides it in half), the Center of Mass lies in that plane
Moment of Inertia for a Rigid Body
I = int[ (r perpendicular)^2 dm]
I = sum(mi ri^2)
Moment of Inertia
(I) rotational counterpart to mass (indicates an object’s inertia to being rotated–called “rotational inertia”)
Newton’s 2nd Law for Rotation
tau = (Iz) alpha
Rotational Equations generated by substituting analogous quantities
x theta v omega a alpha m I p L F tau p = mv L = I omega F = ma tau = I alpha T = 1/2 m v^2 Trot = 1/2 I omega^2
Moment of Inertia for Specific Rigid Bodies
Uniform thin rod rotated about its center of mass:
I = 1/12 m L^2
Uniform thin ring (or cylindrical shell) rotated about its cm:
I = m R^2
Uniform Disk (or cylinder) rotated about its cm: I = 1/2 m R^2
Parallel-Axis Theorem
a rigid body’s movement of inertia (I) about any axis is equal to “I” about a parallel-axis through its cm plus the product of the body’s mass (m) with the square of the distance (L) between the two axes
I = Icm + m L^2
Physical (or Compound) Pendulum
a rigid body that is free to rotate about a horizontal axis under its own weight
T = 1/f = 2 pi sqrt(I/(mgL))
Laminar Motion
all particles in a rigid body move parallel to a fixed plane
Critical Value
(mu crit) the minimum value of coefficient of static friction (mu s) that allows rolling motion with no slipping
Moments of Inertia
Moment of Inertia about x-axis:
Ixx = int[ (y^2 + z^2) dm]
Moment of Inertia about y-axis:
Iyy = int[ (x^2 + z^2) dm]
Moment of Inertia about z-axis:
Izz = int[ (y^2 + x^2) dm]
Products of Inertia
xy Product of Inertia
Ixy = Iyx = - int(x y dm)
xz Product of Inertia
Ixz = Izx = - int(x z dm)
zy Product of Inertia
Izy = Iyz = - int(z y dm)
Moment of Inertia Tensor
9-component mathematical quantity that contains the moments and products of inertia for a rigid body. It is often written as a 3x3 matrix
I = | Ixx Ixy Ixz |
| Iyx Iyy Iyz |
| Izx Izy Izz |
Matrix Multiplication
each matrix entry (Mij) is obtained by multiplying corresponding entries in a Row “i” and Column “j” and adding these products together
Moment of Inertia Equation
I = Ixx cos^2 (alpha) + Iyy cos^2 (beta) + Izz cos^2 (gamma) + 2 Iyz cos(beta) cos(gamma) + 2 Izx cos(gamma) cos(alpha) + 2 Ixy cos(alpha) cos(beta)
I = (n with a ~ ) I n –Note that right side are all matrices
Angular Momentum Equation
L = I dot omega
L = I omega –Note that right side are both matrices
Kinetic Energy Equation
T = 1/2 I omega^2
T = 1/2 (omega with a ~ ) I omega –Note that right side are all matrices
Principal Axes of a Rigid Body
a set of 3 coordinate axes for which the “products of inertia” are zero
-found by diagonalizing the moment of inertia tensor
0 = | Ixx - Ii Ixy Ixz |
| Iyx Iyy - Ii Iyz |
| Izx Izy Izz - Ii |
