Kinematics
A. Description of the Movement
Scientific Terms: The mass begins its travel above where the force of the spring would equal the force of gravity. This causes gravity to pull down on the mass until a point where the spring's forces would slowly cancel the motion gravity is causing. Suddenly, the mass is being pulled back up by the force of the spring until gravity outweighs the spring's forces. This becomes a reoccurring pattern of up and down. Since at each period (the time it takes for the mass to return to its current position after one oscillation) the distance it travels isn't as much as it was for the period before, the mass would in some time return to complete equilibrium (or no net movement) as the period shortens each time until it comes close enough to reaching zero.
Ordinary Terms: The object travels up and down going less far each time. Each time it goes down and back up to its current position, this could be called a period. As time goes on, and the periods become shorter and shorter because the mass is going a smaller distance each time, the mass begins to focus on one point in the middle where gravity pulls down on it just enough to balance the power of the spring. This eventually leads to the mass being on that point with no more movement.
Ordinary Terms: The object travels up and down going less far each time. Each time it goes down and back up to its current position, this could be called a period. As time goes on, and the periods become shorter and shorter because the mass is going a smaller distance each time, the mass begins to focus on one point in the middle where gravity pulls down on it just enough to balance the power of the spring. This eventually leads to the mass being on that point with no more movement.
B. Graphs and Explaination
![Picture](/uploads/3/8/8/1/38816875/70724.jpg?325)
The graph to the right depicts the position vs. time and shows us the motion that takes place during the first period. The object begins going down but not at a linear level. It begins with a steep slope but curves into its minimum point indicating that its velocity is definitely not constant when it goes down. It goes back up the same way it came down, just backwards, this time starting slow and gaining speed. Towards the end we can see it slow again to show its reoccurring pattern.
![Picture](/uploads/3/8/8/1/38816875/6740149_orig.jpg)
The graph to right showing Velocity over time also shows us that the velocity is never constant any point in time. The negative velocity means that the object is going down and the same goes vice versa. The velocity is slowing down too, showing the acceleration isn't constant anywhere within the period.
C. Spring Constant
The spring constant is a term coined to show the stiffness of the spring or in a sense, the flexibility the spring has that the spring has which make the mass move the way it does. It can be solved with this equation where T = time it takes for one period, m = mass, and k = the spring constant :
T = 2pi√(m/k)
0.8 = 2(3.14)√(0.5/k)
0.8/(2 x 3.14) = √(0.5/k)
(0.127388)² = √(0.5/k)²
0.016227 = 0.5/k
0.016227k = 0.5
k = 30.81125 N/m
0.8 = 2(3.14)√(0.5/k)
0.8/(2 x 3.14) = √(0.5/k)
(0.127388)² = √(0.5/k)²
0.016227 = 0.5/k
0.016227k = 0.5
k = 30.81125 N/m
Forces
A. Free Body Diagrams
Equilibrium:
The forces applied at equilibrium can be found a lot easier than the other forces because by definition we know that at at equilibrium, the forces add to zero. Applying Newton's Second Law to this, we can do some rearranging of the force equation to achieve a method of finding what the Spring force is, Force (total) = Force (spring) - force (gravity) Force (total) + Force (gravity) = Force (spring) We can find both, the force of gravity and the spring using Newton's second law and our acceleration graph above. We know equilibrium shows the small part where the spring is at constant velocity, and that makes acceleration 0. F = m x a Force (total) = 0.5 x 0 F = 0 The acceleration gravity has is 9.8 m/s² and this aids us in finding the force of gravity as well as the spring. F (gravity) = 0.5 x 9.8 F = 4.9 Substituting both values into the equation below gives us, Force (total) + Force (gravity) = Force (spring) 0+4.9 = Spring Force = 4.9 N |
Highest Point:
The highest point has the force of gravity stronger than the spring because the mass is about to go down once again. We can figure out what the force of the spring and the total will be using the same method as before, as well as the same graphs/formulas. Force (total) = Force (spring) - force (gravity) Force (total) + Force (gravity) = Force (spring) Finding the acceleration by matching the highest point from the position vs time graph to the same time frame in the acceleration graph, we get F (total) = 0.5 x -1.75 F = -0.875 The force of gravity will stay the same throughout, so we have 4.9 N for that variable. This means we can solve for the spring force. -0.875 + 4.9 = 4.025 N = Force of the Spring |
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Lowest Point:
The forces applied to the lowest point would be gravity and the spring like the others, but this point has the spring's force outweighing gravity's force. We can find out how much by reapplying the previous methods. Force (total) = Force (spring) - force (gravity) Force (total) + Force (gravity) = Force (spring) Matching the lowest point from the position vs time graph to the same time frame in the acceleration graph, we get Force(total) = 0.5 x 1.82 F = 0.91 N Substituting the value of the total as well as gravity (4.9 N) we solve for the spring below. 0.91 + 4.9 = Spring Force 5.81 N = F (Spring) |
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C. Force vs Time Graph
Energy
A. Graphs
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1. The graph displayed by the number 1 shows the Kinetic Energy over time. The movement of the mass can be shown through this graph, the areas of high kinetic energy representing when the mass has it peak acceleration, a little before begins to decelerate and slowly come to a stop at 0.4 seconds. It will then go back up and repeat the same process as we have seen.
2. The Gravitational Potential energy graph is shaped like this because as the height decreases exponentially like the position graph shows it does, the amount of potential energy will decrease as well, in the same way position does . This can be seen with the formula, GPE = (mass)(gravitational acceleration)(height) If height decreases at an exponential rate, then so will PE due to its direct relation. 3. Total Mechanical energy is shaped like this simply because it is a combination of GPE and Kinetic energy. It does give us the insight of how the energy isn't conserved by showing us how at the end of the graph, ME doesn't go as low at it used to, which tells us that energy is being transferred into something else. |
The graph above depicts the Kinetic Energy for the entire motion. The formula for kinetic energy is (1/2)mv² and from this we can identify what is taking place and why. We know the mass would stay the same throughout, but the velocity wouldn't, and squaring it at every level would cause the changes taking place to seem bigger than they actually are. This makes kinetic energy to have such steep increases and and falls.