Problem 1


A 0.5000-kg cart resting on an air track oscillates as shown in the animation (position in meters and time in seconds). NO FRICTION !


What is the spring constant k of the spring? Start
[REQUIREMENT: Use conservation of energy. Set the maximum potential energy equal to the maximum kinetic energy.
You can get the maximum  KE from an approximation of the speed at x = 0 by stepping through the origin with an increment on either side to get  the change in position and using other pertinent information.]  


In Chapter  14, we will discover the position x may be written under special initial conditions  as x = A*cosωt,  where A is the amplitude or magnitude of the maximum displacement from x = 0 and ω is the angular frequency. The expression means the mass on the spring was released from rest at initial x = A.  You may verify this by finding the velocity. Differentiate the formula for x. From this information,  answer and explain the following question. The method of computation in #1 will produce a systematic error such that the calculated k above  will  differ from the  true value. Will computed k be larger or smaller than the true value?  Explain graphically with a sketch.


What is the value of  positive x such that the kinetic energy equals the potential energy ? Start
Find this using a calculation.  How does your  x calculation compare with the x-value inferred from the graph?

Reference:  CH. 6, 7