Lecture 24

Readings

http://umdberg.pbworks.com/w/page/84242443/The%20online%20pendulum%20lab

http://umdberg.pbworks.com/w/page/106891275/Example%3A%20Oscillator%20graphs

http://umdberg.pbworks.com/w/page/106869540/Example%3A%20Oscillator%20calculations

Oscillator Graphs

• Note that total $E$$E$ ( KE plus PE ) is always constant ( if no friction )

Oscillator Calculations

• Consider a projectile of mass "$m$$m$" ( bullet , sticky gum ) moving horizontally toward mass on spring, collides and sticks to block of mass $M$$M$

• Two steps:

  • Step 1 - collision , conservation of momentum

  

  $$Momentum({\rho }_{initial})=m\ast velocit{y}_{initial}$$
  $${\rho }_{final}=(m+M)\ast Velocity$$
  $$\begin{array}{c}{\rho }_{initial}={\rho }_{final}\\ \therefore m\ast velocity=(m+M)\ast Velocity\end{array}$$
  $$Velocity=\frac{m\ast v}{m+M}$$

  • There was an assumption here

    • The collision happened so fast that no oscillation occurred yet

  

  • Step 2 - oscillation , find $A$$A$ ( amplitude of oscillation )

    • I'm going to solve this differently than the reading.

      • I'm going to use conservation of energy

    • Note: now "init" just revers to after the collision

      • This is different than the "init" in Step 1

    • Initially, $M$$M$ ( and now $m$$m$ ) are at equilibrium length of spring, but with speed $V$$V$

• Eventually, spring will be maximally compressed

• Useful Finding from Reading:

  • If $x(t)=A\ast sin({\omega }_0\ast t)$$x(t) = A * sin(\omega_0 * t)$

    • Then , $v(t)={\omega }_0\ast A\ast cos({\omega }_0\ast t)$$v(t) = \omega_0 * A * cos(\omega_0 * t)$

      • This ${\omega }_0\ast A$$\omega_0 * A$ is equal to the maximum velocity , since largest value cosine can have is 1

  • So if you know maximum amplitude, $A$$A$ , then ${\omega }_0\ast A$$\omega_0 * A$ is the maximum velocity

  • Similarly, if you know the maximum velocity, $V$$V$ , then $\frac{V}{{\omega }_0}$$\frac{V}{\omega_0}$ is the maximum amplitude

Clicker Questions