Units of N/m m 2 = N m = J  Total potential energy is Example: Problem A block (m = 1.7 kg) and a spring (k = 310 N/m) are on a frictionless incline (  = 30°). The spring is compressed by x 0 = 0.31 m relative to its unstretched position at x = 0 and then released. What is the speed of the block when the spring is still compressed by x f = 0.14 m?

 x0x0 xfxf x=0 Given: m=1.7 kg, k=310 N/m,  =30°, x 0 =0.31 m, x f =0.14 m, frictionless Method: no friction, so we can use conservation of energy (no rotation) Initially

Finally

Interesting to plot the potential energies xfxf xfxf xfxf x0x0 x0x0 x0x0 Energy PE g PE e PE total E KE 

The Simple Pendulum  An application of Simple Harmonic Motion  A mass m at the end of a massless rod of length L  There is a restoring force which acts to restore the mass to  =0  Compare to the spring F=-kx  The pendulum does not display SHM m  T mg mgsin  L

 But for very small  (rad), we can make the approximation (  <0.5 rad or about 25°)  simple pendulum approximation Arc length Looks like spring force Like the spring constant This is SHM  Now, consider the angular frequency of the spring

Simple pendulum angular frequency Simple pendulum frequency  With this , the same equations expressing the displacement (s), v, and a for the spring can be used for the simple pendulum, as long as  is small  For  large, the SHM equations (in terms of sin and cos) are no longer valid  more complicated functions are needed (which we will not consider)  A pendulum does not have to be a point-particle

The Physical Pendulum  A rigid body can also be a pendulum  The simple pendulum has a moment of inertia  Rewrite  in terms of I  L is the distance from the rotation axis to the center of gravity cg mg L m

Example Use a thin disk for a simple physical pendulum with rotation axis at the rim. a) find its period of oscillation and b) the length of an equivalent simple pendulum. Solution: a)From table 9.1 But we need I at the rim, so apply parallel axis theorem, h=R R M

Since physical pendulum frequency is Distance from rotation axis to cg: L=R Let R=0.165 m (6.5 inches) Would make a good clock!

Note that the period or frequency of a pendulum does not depend on the mass and would be different on other planets b) For an equivalent simple pendulum, we need the simple and disk pendulums to have the same period See example 10