Work, Energy, Torque

Momentum - Center of Mass Theorem

The linear momentum of a system of particles is equal to the product of the total mass M of the system and the velocity of the center of mass.

Collisions

Inelastic: Momentum is conserved Elastic: Kinetic energy and momentum is conserved

On any path, a force constant in magnitude and direction will do the same amount of work

Equations

Linear Motion

Energy

Work ($J$), as a function of distance ($m$) and force ($N$), where $\theta$ is the angle between distance and force: $W=Fdcos(\theta )$ $W={\int }_A^B\vec{F}\cdot d\vec{l}$ Linear kinetic energy, $K$ ($J$) as a function of mass ($kg$) and velocity ($m/s$): $K=0.5m{v}^2$ Above, rewritten for velocity in terms of energy and mass: $v=\sqrt{\frac{2K}{m}}$ Gravitational Potential energy, U, (J) in terms of mass, gravitational constant, and height $\Delta U=mgh$

Only non conservative forces can change the total energy of an object or system: $W_{nc}=U_i+K_i-U_f-K_f$ Force in each direction is equal to the change in potential energy over the change in position in that direction: $F_x=\frac{\delta U}{\delta x}$ $F_y=\frac{\delta U}{\delta y}$ $F_z=\frac{\delta U}{\delta z}$ The disassociation energy of a system is it’s final potential energy less it’s origin potential energy: $U_{dis}=U_{\infty }-U_r$

Momentum

$P$ = momentum in $kgm/s$ $m$ = mass in kg $v$ = velocity in m/s $P=mv$ $\Delta P=m\Delta v$ $\Delta P=J$ Impulse (newton seconds) = $J$ = impulse in $kgm/s$ $F$ = force in N $J=Ft$ $J=m\Delta v$

Angular Motion

Moments of Inertia

Parallel Axis Theorum: (when things are rotating not about their center)

The moment of inertia of a thing not rotating about it’s center is equal to it’s moment of inertia about it’s center plus it’s moment of inertia as if it were a particle $I=I_{CM}+M{L}^2$

Finding Torque

Radius and tangential component of force: $\tau =r{F}_{tan}$ Lever arm (perpendicular distance between line of force and rotation center) and Force: $\tau =lF$ Vector cross product of torque and point of action: $\vec{\tau }=\vec{r}\vec{F}$

Energy

Rotational kinetic energy, $K$ (J) in terms of angular velocity, $\omega$, (rad/s ) and moment of inertia, $I$ ($kg{m}^2$ but not $J$): $K=\frac{1}{2}I{\omega }^2$ $I=\frac{2K}{{\omega }^2}$ Torque as a function of angular acceleration ($rad/s^2$) and moment of inertia: $\Sigma \tau =I\alpha$

Momentum

Angular momentum (L) as a function of moment of inertia and angular velocity $\omega$ $L=I\omega$ Torque is the change in angular momentum: $\tau =\frac{\Delta L}{\Delta t}$

Precession

Angular velocity of precession ${\omega }_P$ is magnitude of torque on gyroscope over angular momentum of gyroscope ${\omega }_P=\frac{|\tau |}{|L|}$