Students will learn to calculate periods, frequencies, etc. of spring systems in harmonic motion.
Key Equations
Guidance
 In harmonic motion there is always a restorative force, which attempts to restore the oscillating object to its equilibrium position. The restorative force changes during an oscillation and depends on the position of the object. In a spring the force is given by Hooke’s Law:
F=−kx
 The period,
T , is the amount of time needed for the harmonic motion to repeat itself, or for the object to go one full cycle. In SHM,T is the time it takes the object to return to its exact starting point and starting direction.
 The frequency,
f, is the number of cycles an object goes through in1 second. Frequency is measured in Hertz(Hz) .1 Hz=1 cycle per sec.
Example 1
Watch this Explanation
Simulation
Mass & Springs (PhET Simulation)
Time for Practice
 A rope can be considered as a spring with a very high spring constant
k, so high, in fact, that you don’t notice the rope stretch at all before it “pulls back.” What is the
k of a rope that stretches by1mm when a100kg weight hangs from it?  If a boy of
50kg hangs from the rope, how far will it stretch?  If the boy kicks himself up a bit, and then is bouncing up and down ever so slightly, what is his frequency of oscillation? Would he notice this oscillation? If so, how? If not, why not?
 What is the
 If a
5.0kg mass attached to a spring oscillates 4.0 times every second, what is the spring constantk of the spring?  A horizontal spring attached to the wall is attached to a block of wood on the other end. All this is sitting on a frictionless surface. The spring is compressed
0.3m . Due to the compression there is5.0J of energy stored in the spring. The spring is then released. The block of wood experiences a maximum speed of25m/s . Find the value of the spring constant.
 Find the mass of the block of wood.
 What is the equation that describes the position of the mass?
 What is the equation that describes the speed of the mass?
 Draw three complete cycles of the block’s oscillatory motion on an
x vs.t graph.
 A spider of
0.5g walks to the middle of her web. The web sinks by1.0mm due to her weight. You may assume the mass of the web is negligible. If a small burst of wind sets her in motion, with what frequency will she oscillate?
 How many times will she go up and down in one s? In
20s ?  How long is each cycle?
 Draw the
x vst graph of three cycles, assuming the spider is at its highest point in the cycle att=0s .
Answers to Selected Problems
 a.
9.8×105N/m b.0.5mm c.22Hz 
3.2×103N/m  a.
110N/m d.v(t)=(25)cos(83t)  a.
16Hz b.16 complete cycles but32 times up and down,315 complete cycles but630 times up and down c.0.063s
Investigation
 Your task: Match the period of the circular motion system with that of the spring system. You are only allowed to change the velocity involved in the circular motion system. Consider the effective distance between the block and the pivot to be to be fixed at 1m. The spring constant(13.5N/m) is also fixed. You should view the charts to check whether you have succeeded. Instructions: To alter the velocity, simply click on the Select Tool, and select the pivot . The Position tab below will allow you to numerically adjust the rotational speed using the Motor field. To view the graphs of their respective motion in order to determine if they are in sync, click on Chart tab below.

 Now the mass on the spring has been replaced by a mass that is twice the rotating mass. Also, the distance between the rotating mass and the pivot has been changed to 1.5 m. What velocity will keep the period the same now?
