Lecture 23

Readings

http://umdberg.pbworks.com/w/page/51902440/Hanging%20mass%20on%20a%20spring

http://umdberg.pbworks.com/w/page/124452119/Workout%3A%20Hanging%20mass%20on%20a%20spring

http://umdberg.pbworks.com/w/page/64893635/The%20pendulum%20%282013%29

Hanging Mass on Spring

• Motion is the same as for horizontal mass attached to spring without friction
• The confusion is "what is equilibrium length?"

• The mass $m$$m$ stretches spring by $x_S$$x_S$
• How much stretch? Use $F=-k\ast x$$F = -k * x$
• At new equilibrium:

• Forces balance so: $mass\ast gravity=k\ast x_S$$mass * gravity = k * x_S$
• $\therefore x_S=\frac{m\ast g}{k}$$\therefore x_S = \frac{m*g}{k}$
• But main point, all oscillation occurs about new equilibirum
• No forces are exerted on mass at new equilibrium
• If we choose coordinate system:

• Motion = $y(t)=A\ast cos(({\omega }_0\ast t)-\psi )$$y(t) = A * cos((\omega_0*t)-\psi)$

Pendulum

• Using energy approach, and assuming oscillation angle is small, we can derive:

• $E_0$$E_0$ is the unchanging total energy
• $\frac{mass\ast velocit{y}^2}{2}$$\frac{mass * velocity^2}{2}$ is kinetic energy
• $\frac{mass\ast gravity\ast Length\ast {\theta }^2}{2}$$\frac{mass *gravity * Length * \theta^2}{2}$is potential energy ( from $m\ast g\ast h$$m*g*h$ )

• Which is exactly the same as horizontal or vertical mass on spring
• So motion is the same, Simple Harmonic Motion ( SHM )

• Frequency of oscillation:

• Note that $\sqrt{\frac{gravity}{Length}}$$\sqrt{\frac{gravity}{Length}}$ deoesn't depend on mass.

  • It only depends on Length and planetary gravity

• Angle $\theta$$\theta$ is in radians