mech final

what is the equation for the center of mass?

int (cos(x)sin(x))dx

(1/2)sin²(x)

arc length formula

s = r(theta)

line integral for 1D com

(1/L) ∫ r ds

velocity cross product

angular momentum L

SHM

moment of inertia

(mass)*(distance from axis of rotation)²

torque is defined as

the perpendicular distance from the lever

torque formula

T = r x F

the period of a simple pendulum

mass density

for 3D solids

$$\varrho=\frac{M}{V}$$

surface density

for 2D surfaces

$$\sigma=\frac{M}{A}$$

line density

for 1D objects

$$\lambda=\frac{M}{L}$$

moment of inertia, integral form

$$I=\int\rho^2dm=\int\rho^2\varrho dV$$

rotational kinetic energy

$$T_{rot}=\frac12I\omega^2$$

moment of inertia tensor

characteristics of the principal axis

• L and w parallel

• diagonal inertia tensor I

• moments of inertia called principal moments (diagonal elements of I)

• axis through center of mass is always a principal axis

inertia tensor element for an object with continuous mass

$$I_{\times\times}=\int\left(y^2+z^2\right)dm=\int\left(y^2+z^2\right)\varrho dV$$

$$I_{xy}=\int xydm=\int xy\varrho dV$$

what is w for rotation about a cube’s diagonal?

w=<1,1,1>

the inertia tensor I of rotation about a point translated from the center of mass

$$\Delta=\left(c.o.m.\right)-\left(a,b,c\right)=\left(A,B,C\right)$$

$$I_{xy}=I_{xy}^{COM}-MAB$$

$$I_{xx}=I_{xx}^{COM}+M\left(B^2+C^2\right)$$

BAC-CAB rule

$$A\times\left(B\times C\right)=B\left(A\cdot C\right)-C\left(A\cdot B\right)$$

kinetic energy of a particle

$$T=\frac12v\cdot p$$

kinetic energy from angular momentum