# Forces in Equilibrium

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Lab III, Problem 2: Forces in Equilibrium

Zachery Smith

November 16th, 2017

Physics 1301W, Professor: M. Marshak, TA: C. Schimming

Abstract

The idea of mechanical energy stored in springs and a springs behavior when acted on by an outside force were studied. A video of a spring with a given mass attached was taken and the system’s motion was analyzed. In this analysis, it was found that the mechanical energy stored in springs is directly proportional to the square of the spring’s displacement from equilibrium.

Introduction

While designing the suspension for any type of modern car, springs are regularly used to dissipate the energy of the car bouncing up and down. In general, a spring can gain internal energy by being both stretched and compressed from its equilibrium position; but, there are several different variables, like the spring’s stiffness (spring constant) and the length the spring is stretched from equilibrium, that can affect the amount of mechanical energy stored in the spring at any given time. The lab group, challenged with the task of designing a suspension system for a new sports car, must understand the storage of energy within springs before being able to assess how well certain springs will function as energy dissipates while the car is in motion.

Prediction

Using general knowledge and physics formulas and concepts, the behavior of the spring and its relationship to the storage of energy can be predicted. First off, springs can be significantly stretchy or stiff. This is due to an aspect known as the spring constant (k), which is affected by several characteristics of each unique spring’s makeup: material used, thickness of the wire, diameter of coils, and even the number of coils per unit length. From simply handling any spring, it's noticed that when one is compressed or stretched by an external force from its equilibrium position (the length of the spring in the presence of no external forces), the spring pushes or pulls in the opposite direction with an equal force. The amount of force (F) the spring applies varies directly with the distance it’s stretched (Δx), and can be shown in the following equation:

Equation #1

F = (k) x (Δx)

Once the external force is no longer applied, the natural response of the spring is to snap back toward equilibrium. It can also be noted that it requires a substantial amount of additional force to compress or stretch a spring as its distance from its equilibrium position increases, which means the spring has more internal energy (U) allowing it to push back with more force as Δx increases. This relationship can be shown through the equation -

Equation #2

U = (½) x (k) x (Δx)2

Given this, and the fact that the force applied by the spring increases as Δx increases, it was predicted that the internal energy of a spring is directly related to some degree of position greater than one (in particular, to the 2nd degree).

In order to accurately obtain values for these internal energies of the spring at different times, the total mechanical energy (kinetic + gravitational potential) of the mass can be subtracted from the total energy of the system (the initial amount of energy before motion begins). The kinetic and potential energies can be found at any time using the following equations with these variables: m = mass of the mass, vt = the instantaneous velocity of the mass at time t, g = gravitational acceleration, and ht = the height at time t relative to the origin.

Equation #3 & #4

Potential energy = (m) x (g) x (ht)                             Kinetic energy = (½) x (m) x (vt)2

Equation #5

Mechanical energy = Kinetic energy + Potential energy

Procedure

A video recorder was placed a distance away from the plane of motion of the spring in order to accurately gather data pertaining to the position of the spring throughout its motion. A meter stick was also placed in the plane of motion so that a length reference was present for later analysis on the computer. A spring with an unknown spring constant was then hung over the edge of a table so that a mass could be hung from it. This was done to ensure that there would be solely forces in the vertical direction acting on the spring. The mass was then weighed using a balance with an uncertainty of ± .0005 kg. At this point, the apparatus was ready for testing. The video recording began, the mass was placed on the free hanging hook of the spring and held at the initial equilibrium position of the spring, and then the mass was dropped. A recording of at least two full cycles of the spring’s motion (oscillations) was needed, and then using Motionlab data analysis software, the video was analyzed. While the mass was in motion both upward and downward, points were placed on a fixed location of the end of the spring at various times, which provided its change in position (relative to an origin placed at its starting point) over the time of its motion. With this data, the velocity of the mass at each instance could be calculated instantaneously, which would provide the values needed to calculate the energy (both potential and kinetic) at any given point throughout the mass’s motion. A video recorder was placed a distance away from the plane of motion of the spring in order to accurately gather data pertaining to the position of the spring throughout its motion. A meter stick was also placed in the plane of motion so that a length reference was present for later analysis on the computer. A spring with an unknown spring constant was then hung over the edge of a table so that a mass could be hung from it. This was done to ensure that there would be solely forces in the vertical direction acting on the spring. The mass was then weighed using a balance with an uncertainty of ± .0005 kg. At this point, the apparatus was ready for testing. The video recording began, the mass was placed on the free hanging hook of the spring and held at the initial equilibrium position of the spring, and then the mass was dropped. A recording of at least two full cycles of the spring’s motion (oscillations) was needed, and then using Motionlab data analysis software, the video was analyzed. While the mass was in motion both upward and downward, points were placed on a fixed location of the end of the spring at various times, which provided its change in position (relative to an origin placed at its starting point) over the time of its motion. With this data, the velocity of the mass at each instance could be calculated instantaneously, which would provide the values needed to calculate the energy (both potential and kinetic) at any given point throughout the mass’s motion. Note: Housing & Residential Life reserves the right to inspect resident rooms/apartments within 30 days prior to the end of a term to assess maintenance/repair needs. Housing & Residential Life may do these checks throughout a semester and will give the resident a non-emergency 24-hour notice.

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