Activity 2- Forces Oscillating Systems

Yong-Nan Chang, Brooklyn Morrill, Ronan Enright, Fernando Varini Activity 2: Forces and Oscillating Systems Background Material An oscillating system is one that undergoes periodic motion. Such a system cannot have a constant net force, since periodic motion necessarily involves changing acceleration (changing in direction and/or magnitude). Consider as a prototype a mass suspended from a simple spring. There are two forces acting on the mass: a gravitational force downwards and a force upwards from the spring. We assume the force exerted by the spring obeys Hooke’s Law: F =− ky (1) The net force is given by F net = F s + F g = k Δ y − Δmg = ma (2) This net force is always in a direction back towards the equilibrium position where F net = 0. This is a key part of an oscillating system, that it is subject to a restorative force, that always tends to return it to equilibrium. If the mass is just hanging, then F net = 0 is true, and equation 2 can be written as k  y = Δmg (3) We will study two cases of oscillatory motion today: a simple pendulum, and a mass suspended from a spring. For the case of the simple pendulum, a theoretical analysis indicates that, for small angles of oscillation, the period of the pendulum motion depends only on the length: T = 2 π √ l g →T 2 = 4 π 2 g l (4) For the case of a mass suspended from a spring and oscillating vertically, the motion can be described as a sine function, y = A sin ( ωt + φ ) , (5) where φ is a phase angle, that relates to the position at time t = 0. The period of the motion can be calculated as T = 2 π √ m k →T 2 = 4 π 2 k m (6) where k is the spring constant described in Hooke’s Law. It may seem that a simple pendulum and a mass attached to a spring are rather contrived examples of periodic motion, that don’t reflect real situations. However, many systems behave in ways that can be modeled very accurately with the simple ideas we have developed here. For example, a building or tower might sway in the wind or during an earthquake, or the floor vibrates when you walk across it or a truck drives by. Any system that experiences vibrations of any sort (which is virtually every system) can be modeled quite well by using the ideas of Hooke’s Law or oscillations similar to a pendulum.

2. What is your theoretical slope? 4.02841 3. Compare to the theoretical result above. In particular, compare to your measured slope: The theoretical slope is only 0.2686 off the measured slope, what is pretty close but not exact. C. Further investigations 1. Verify that period is independent of mass. Change the mass and note the period of each. (try to start at the same initial angle) Trial mass Period 1 1 1.45 2 1.5 1.45 2. Verify that the period has at most a weak dependence on the angle through which the pendulum oscillates. Trial angle Period 1 90 1.98 2 60 1.8 3 5 1.68 4 30 1.71 II. Study a mass suspended from a spring. https://phet.colorado.edu/en/simulation/masses- and-springs Choose the lab option, and turn on the Natural line and the mass equilibrium line. Also pull out the ruler, and slide the spring constant1 to either small or large

A. Measure the spring constant. Place the mass on the spring, and note where the equilibrium mark is. Make sure to measure off the “natural line”, and then the mass amount Trial Mass  Mass (subtract the two) Distance from Natural line  Length (subtract last two) 1 100gr 25cm 2 150gr 50gr 34.5cm 10cm 3 200 50gr 44.5cm 10cm 4 300 100gr 62.5 28cm Ave  Mass 66.67 Ave  Length 16 1. k  y = Δmg 2. Calculate the average change in mass and the average change in length and record it on the bottom of the table. 3. What is the spring constant? (use equation (3) above) 40.835 4. Turn off the damping, pull out the timer, and turn on the period trace. 5. For several different masses, record the period of oscillation. Trial Mass Period Period 2 1 300gr 1.4s 1.96s 2 2 200gr 1.17s 1.369s 2 3 150gr 1.02s 1.04s 2 4 100gr 0.8s 0.64 2 6. 7. Linearize a plot comparing period and mass, similar to the pendulum by graphing T 2 = 4 π 2 k m and verify the theoretical by using your spring data from above, and find the measured slope in Excel.