PHYS 218 Exam 3

Energy and Momentum

Momentum: $P=mv$ Impulse: $J=\int _{t_{i}}^{t_{f}}F\ \mathrm {d} t=P_{f}-P_{i}$

Collisions

Conservation of Momentum

Elastic: (All energy conserved: $K_{i}=K_{f}$)

Inelastic: (Max loss of energy: $K_{f}=K_{i}$) When two bodies stick together.

Center of Mass

Position: $r_{cm}={\frac {\sum m_{i}r_{i}}{\sum m_{i}}}$

Velocity: $v_{cm}={\frac {\sum m_{i}v_{i}}{\sum m_{i}}}$

Acceleration: $a_{cm}={\frac {\sum m_{i}a_{i}}{\sum m_{i}}}$

In a system

Sum of all external forces is the total mass × a_cm = ${\tfrac {dP_{cm}}{dt}}$

If sum of forces in any component direction is 0, the energy is conserved in that direction

Rotational Kinematics

${\begin{aligned}\omega &={\frac {\mathrm {d} \theta }{\mathrm {d} t}}\\\alpha &={\frac {\mathrm {d} \omega }{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\\\omega &=\omega _{0}+\alpha t\\\theta -\theta _{0}&=\omega _{0}t+{\frac {1}{2}}\alpha t^{2}\\v&=R\omega \\a_{T}&=R\alpha \\a_{R}&=R\omega ^{2}\end{aligned}}$

Moment of Inertia

With respect to an axis Q:

$I_{Q}=\sum m_{i}r_{iQ}^{2}$

Parallel Axis Theorem

Rotate an object (Moment of Inertia for center of mass is known) around a new axis at a point Q)

$I_{Q}=I_{cm}+Md^{2}$

Angular Momentum

Conversion from linear Momentum: Sum of the cross product of position and linear momentum for each particle

$L_{Q}=\sum r_{iQ}\times m_{i}v_{i}$

Object rotating around its center of mass:

$L=I\omega$

Rotational Energy

$K={\frac {1}{2}}I\omega ^{2}$

Torque

Cross product between position applied and force:

$\tau =r\times F$

Sum of all torques is equivalent to moment of inertia times the rotational acceleration.

$\sum \tau =I\alpha ={\frac {\mathrm {d} L}{\mathrm {d} t}}$

If the sum of all torques is equal to 0, then we have conservation of angular momentum.

Precession

occurs when angular momentum L is perpendicular to a torque. A rotating object will move around an axis parallel to the applied force:

$\Omega ={\frac {\tau }{L}}$

Topics for Exam

1. Angular Momentum
   • When it is conserved an when it is not
   • Conservation of momentum and parallel axis theorem
2. Rotational Kinematics (falling yo-yo)
   • Use conservation of energy
   • Use kinematics equations (sum of forces, sum of torques)
3. Equilibrium Problems
   • Find tension (make sure that sum of forces and sum of torques are zero from any point)
4. Energy and Collisions
   • Is it elastic or inelastic? Why?