Partner: Rida (3rd Period)
Spring and Mass
Kinematics
Kinematics is a type of mechanics that deals with the motion of objects. In this lab, a spring was hung from a ring stand and a mass of 0.5 kilograms was attached to it. The mass immediately pulled down the spring, but the spring quickly shot back up. This pattern of motion continued repeatedly. Over time the mass-spring system became slower, and dropped and shot up less.
This graph shows the y-position (height) of the mass as the spring stretches and compresses. From the graph we can immediately see that it travels in a harmonic motion. It starts out at its highest point and over the course of about 0.33 seconds, the mass' height decreases. At this point, it is at it's lowest point since the spring is fully stretched. Starting from about 0.33 seconds to 0.77 seconds, the mass' height increases. At 0.77 seconds, the mass is once again at it's highest point. We see that the period of the position vs. time graph of this system is about 0.77 seconds.
This graph shows the velocity of the mass-spring system as the spring stretches and compresses. From the graph we can see that it is also roughly a harmonic motion. The lowest point of the system is around 0.33 seconds with a velocity of about 0 m/s. The highest point of the system is around 0.77 seconds with a velocity of about 0 m/s as well. These values are because at the highest and lowest points, the mass-spring system stops for a moment in order to change direction to start moving down and up respectively, resulting in a velocity of zero at both points. The equilibrium point is around 0.56 seconds and corresponds to the greatest velocity on the graph of about 0.3 m/s. The equilibrium point will correspond to both the greatest and least velocity values of the graph depending on which way the system is moving when it passes through the point. If the system is moving downwards, the velocity is positive and if the system is moving upwards, the velocity is negative. Since the system is moving downwards around 0.57 seconds, the velocity is positive.
This graph shows the acceleration of the mass as the spring stretches and compresses. From the graph we can see that it is also roughly a harmonic motion. The lowest point of the system is around 0.3 seconds with an acceleration of about 4.0 m/s^2. The highest point of the system is around 0.77 seconds with an acceleration of about -2.1 m/s^2. The equilibrium point is around 0.57 seconds with an acceleration of about 0 m/s^2. The lowest point has a positive acceleration because after that point, the system starts moving upwards and its velocity increases. This means that the acceleration is negative after the highest point as the system begins to move downwards and its velocity decreases. Since the velocity is at its greatest and least values at the equilibrium point, the system neither accelerates or decelerates at that exact point.
The reason why the mass-spring system was described as it was is because springs have elasticity, the property of an object or material that causes it to restore itself back to its original form before it was distorted. The more precisely an object restores itself, the more elastic the object is said to be. When a spring is stretched, it exerts a force that restores it back to its original length. This explains harmonic motion like the motion of the spring and mass.
Spring constant, measured in Newtons per meter, describes the rigidness of a spring as it is compressed and decompressed. It can be defined in the equation to find the period, or the time it takes for one oscillation of the spring, to occur. The formula to find the period is T = 2π * √(m/k), where T is the period is seconds (s), m is the mass of the weight on the spring in kilograms (kg), and k is the spring constant in Newtons per meter (N/m). In order to solve for the spring constant, we can algebraically rearrange the formula for the period. Based on the graphs, it can be determined that the period for the spring used is 0.8 seconds. The weight placed on the spring was 500 grams or 0.5 kilograms.
Spring constant, measured in Newtons per meter, describes the rigidness of a spring as it is compressed and decompressed. It can be defined in the equation to find the period, or the time it takes for one oscillation of the spring, to occur. The formula to find the period is T = 2π * √(m/k), where T is the period is seconds (s), m is the mass of the weight on the spring in kilograms (kg), and k is the spring constant in Newtons per meter (N/m). In order to solve for the spring constant, we can algebraically rearrange the formula for the period. Based on the graphs, it can be determined that the period for the spring used is 0.8 seconds. The weight placed on the spring was 500 grams or 0.5 kilograms.
T = 2π * √(m/k)
T/(2π ) = √(m/k) (T/(2π))^2 = m/k [(T/(2π))^2]*k = m k = m/[(T/(2π))^2] |
k = m/[(T/(2π))^2]
k = 0.5/[(0.8/(2π))^2] k = 0.5/(0.127)^2 k = 31 N/m |
Forces
There are two forces that act upon the mass-spring system. The first force is gravity which pulls the mass-spring system down. The second force is the force of the spring which shoots the mass-spring system up.
Newton's Second Law is described by F = ma where F is the total net force in Newtons (N), m is mass in kilograms (kg), and a is acceleration in meters per second squared (m/s^2). The force of the spring, Fspring, and the force of gravity, Fgravity, together add up to the total net force of the mass-spring system. The mass of the weight, 0.5 kg, is constant at all positions. The Fgravity, defined by the equation of mass X gravitational acceleration (9.8 m/s^2), is therefore also constant at all positions. At the highest position, the the acceleration was -2.1m/s^2, resulting in Fspring being equal to 3.85 N.
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Fspring, and Fgravity, together add up to the total net force of the mass-spring system. The mass of the weight, 0.5 kg, is constant at all positions. The Fgravity, defined by the equation of mass X gravitational acceleration (9.8 m/s^2), is therefore also constant at all positions. At the equilibrium position, the acceleration was 0 s^2, resulting in Fspring being equal to 4.9 N.
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Fspring, and Fgravity, together add up to the total net force of the mass-spring system. The mass of the weight, 0.5 kg, is constant at all positions. The Fgravity, defined by the equation of mass X gravitational acceleration (9.8 m/s^2), is therefore also constant at all positions. At the lowest position, the acceleration was 4.0 m/s^2, resulting in Fspring being equal to 6.9 N.
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This graph shows the force over time as the spring stretches and compresses. From the graph we can immediately see that it travels in a harmonic motion. It starts out with very little force as the mass is in equilibrium. As the spring is stretched though, the spring gains potential energy because the height increases. The lowest point of the system is around 0.33 seconds with a force about 1.0 N. The highest point of the system is around 0.77 seconds with a force of about -0.1 N. The equilibrium point is around 0.56 seconds with a force about 0 N.
Energy
Mechanical energy is defined as the total energy possessed by an object while in motion and at rest. There are two types of mechanical energy, kinetic energy and potential energy. Kinetic energy is the energy that an object possesses while in motion. Potential energy is the stored energy of an object based on the object's position. Therefore, mechanical energy is the sum of kinetic and potential energy.
The lowest point of the system is around 0.33 seconds with a kinetic energy value close to 0 J. The highest point of the system is around 0.77 seconds with a kinetic energy value close to 0 J. The equilibrium point is around 0.56 seconds with a kinetic energy value of about 0.04 J. The equation for kinetic energy is KE = 1/2mv^2, so since the velocity is 0 m/s at both the highest and lowest points, the kinetic energy is 0 J in both cases as well. Since the absolute velocity values are greatest at all the equilibrium points, the equilibrium points have the highest kinetic energy values. Therefore, the graph dips at the beginning of the period until it reaches the lowest position, increases until it reaches the equilibrium point, and decreases until it reaches the highest point.
The lowest point in the system is around 0.33 seconds with a potential energy value of about -0.15 J. The highest point in the system is around 0.77 seconds with a potential energy value of about 0.20 J. The equilibrium point is around 0.56 seconds with a potential energy value of about 0 J. Since the equation for potential energy is PE = mgh, the amount of potential energy decreases as the height of the mass-spring system decreases and the amount of potential energy increases as the height of the system increases. Therefore, the graph dips until the system reaches the lowest point and increases until it reaches its highest point.
The lowest point of the system is around 0.33 seconds with a total mechanical energy value of about -0.15 J. The highest point of the system was around 0.77 seconds with a total mechanical energy value of about 0.20 J. The equilibrium point was around 0.56 seconds with a total mechanical energy value of about 0.04 J. Total Mechanical Energy is the sum of potential and kinetic energy. Therefore, the graph should be a straight line as the total amount of energy in the system should be consistent. However, it can be seen that some energy was lost due to the dip in the graph. According to the Law of Conservation of Energy, energy can neither be created nor destroyed, leading us to assume that some energy was lost and regained from the tension of the spring during the mass-spring system's motion.
![Picture](/uploads/3/8/8/5/38851515/8275898.png?432)
In this graph, the same trend displayed in the kinetic energy graph for one period is repeated every 0.8 seconds, or the time for one period. There are some inconsistencies in the graph at the beginning of the system, but the graph tends to stick to the same trend of dipping at the beginning of each period until it reaches the lowest point, increasing until it reaches the equilibrium point, decreasing until it reaches the highest point, and increasing until it reaches the next equilibrium point. As can be seen by the graph, the peaks at each period tend to get slightly smaller over time and this is due to a loss of energy in the form of heat during the mass-spring system's motion.
Sources
http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy
http://www.physicsclassroom.com/class/energy/Lesson-1/Mechanical-Energy
http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html
http://www.physicsclassroom.com/class/energy/Lesson-1/Mechanical-Energy
http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html