PHYS 218 Exam 3

Energy and Momentum

Momentum: $P=mv$ Impulse: $J=\underset{t_i}{\overset{t_f}{\int }}F\mathrm{d}t=P_f-P_i$

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 = $\frac{d{P}_{cm}}{dt}$

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

Rotational Kinematics

$\begin{array}{rl}\omega & =\frac{\mathrm{d}\theta }{\mathrm{d}t}\\ \alpha & =\frac{\mathrm{d}\omega }{\mathrm{d}t}=\frac{{\mathrm{d}}^2\theta }{\mathrm{d}{t}^2}\\ \omega & ={\omega }_0+\alpha t\\ \theta -{\theta }_0& ={\omega }_{0}t+\frac{1}{2}\alpha t^2\\ v& =R\omega \\ a_T& =R\alpha \\ a_R& =R{\omega }^2\end{array}$

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)

$I_Q=I_{cm}+M{d}^2$

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:

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

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?