length (m) grav (N) 0.057 0.977076 0.116 1.460611 Graph of the spring stretch length as a function of the 0.222 2.450048 gravitational Force on the load mass 0.157 1.947285 3.5 0.285 2.936722 w y = 8.6911x + 0.4995 2.5 .. .. gravitational Force (N) 1.5 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 stretch length (m)linearized x^2 Graph 3 period ^2 mass 0.583696 0.0996 0.793881 0.14889 1.089936 0.1985 1.252161 0.24975 1.495729 0.29936 Linearized Graph of average oscillation period as a function of load mass 0.4 0.35 y = 0.2179x - 0.0281 0.3 0.25 load mass (Kg) 0.2 0.15 T H 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 average oscillatikn ^2 (s)____...__ ..____ .__,___r__._._._ .______._______.__ Q8. Use the theoretical model of a mass-spring system and your graph to determine the value of the spring constant. Calculate the percent (difference or error, as appropriate) between the spring constants from your linearized graph from Q6 and the one you constructed in Q4. Show all work with units. Physics 195: Simple Harmonic Systems Page 1 MS Oscillations, Periodicity and Conservative Forces Objective: To examine the behavior of conservative systems; To relate the inertial and energy storing properties of the system to the period of oscillation; To use oscillation behaviors to determine physical properties of objects. Hypothesis: Systems that 'store' energy deform when a force is applied. Each material should have a unique deformation value. Systems that store energy behave in a periodic fashion, allowing them to be used for timing purposes. Mathematical Models and Reference Values Used: Objects that obey Hooke's Law are simple to analyze, since the amount of force required to deform the object is directly proportional to the amount of deformation. This is written l,,,,-,_._,|=kx_ The value of 'k' is called the spring constant and measures the 'stiffness' of the spring. Since restoring forces lead to periodic behavior, the periodicity of a system can be related to physical characteristics of materials and the type of conservative force acting in the system. When a mass (m) is attached to a spring (k), the mass-spring system will oscillate with an amplitude given by x(t)=xm,,cos[wt+) , where xmax is the amplitude of oscillation, m is the angular frequency of oscillation and CD is the phase of the oscillation. |f oscillation is started at xmax then the phase is zero. The period of oscillation T is related to the angular frequency by ii"=2(77r . Overview By changing load mass and oscillation amplitude, you will investigate how (or if) the period of oscillation is affected. Equipment List: Plumb Bob, Meter stick, Springs, Mass Hanger and Slotted Mass Sets. 1. Planning Phase: Mass-Spring System Q1. Describe the procedure by which you will measure the period of oscillation. What analysis will you perform and how will you deal with standard error? Answer using complete sentences. Q2. How do you expect the period of oscillation to depends on load mass? The load mass refers to the hanger and any masses placed on it. State this relationship as a proportionality, e.g. Team2 or Taxi , etc. Why do you think this is the case? Answer with complete sentences