Free-Body Diagrams:
Objects' Forces at differing Positions
This is the diagram for when the 500 gram mass is at it's highest point during oscillation; it shows how the force of gravity significantly outmatches the force of the spring from this angle.
Over to the left, I have set up Newton's 2nd Law to complement this diagram of the object at it's highest point. The total force is equal to the mass in kilograms times the acceleration in m/s². This is -2.08 Newtons, so keep this aside.
The force of gravity is the mass in kilograms times the rate of gravitational pull in m/s².
The equation of total force equalling the force of the spring minus the force of gravitational pull can be rewritten as the force of gravity equals the total force plus total gravity so that we can solve for the force of the spring.
Adding the total and gravitational forces gives a spring force of 2.82 Newtons, when the object is at it's highest point during a single period of oscillation.
|
\[\large F\scriptsize{total}\space=\space\normalsize m\space\times\space a
\]
\[\large F\scriptsize{total}\space=\space \normalsize 0.5\space\times\space (–4.16)
\]
\[\large F\scriptsize{total}\space=\space\normalsize \space–2.08\space N
\]
\[\large F\scriptsize{G}\space=\space\normalsize m\space\times\space g
\]
\[\large F\scriptsize{G}\space=\space\normalsize 0.5\space\times\space 9.8
\]
\[\large F\scriptsize{G}\space=\space\normalsize 4.9\space N
\]
\[\large F\scriptsize total\space=\space\large F\scriptsize spring \space -\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\large F\scriptsize total \space +\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\large F\scriptsize total \space +\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\normalsize (-2.08)\space +\space \normalsize 4.9
\]
\[\large F\scriptsize spring\space=\space\normalsize 2.82 \space N
\]
|
This illustration of the free body shows that the object is perfectly in sync of up and down (no side-to-side motions AT ALL), and in the middle of a given period (if object was released at the top). If this stands true, then the force of the spring and the force of gravity cancel each other out due to their equivalencies.
Over to the left, I have set up Newton's 2nd Law to complement this diagram of the object in equilibrium. The total force is equal to the mass in kilograms times the acceleration in m/s². This is -0.25 Newtons, so keep this aside.
The force of gravity is the mass in kilograms times the rate of gravitational pull in m/s².
The equation of total force equalling the force of the spring minus the force of gravitational pull can be rewritten as the force of gravity equals the total force plus total gravity so that we can solve for the force of the spring.
Adding the total and gravitational forces gives us the spring force of 4.65 Newtons, when the object is in equilibrium during a period of oscillation.
|
\[\large F\scriptsize{total}\space=\space\normalsize m\space\times\space a
\]
\[\large F\scriptsize{total}\space=\space \normalsize 0.5\space\times\space (–0.5)
\]
\[\large F\scriptsize{total}\space=\space\normalsize \space–0.25\space N
\]
\[\large F\scriptsize{G}\space=\space\normalsize m\space\times\space g
\]
\[\large F\scriptsize{G}\space=\space\normalsize 0.5\space\times\space 9.8
\]
\[\large F\scriptsize{G}\space=\space\normalsize 4.9\space N
\]
\[\large F\scriptsize total\space=\space\large F\scriptsize spring \space -\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\large F\scriptsize total \space +\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\large F\scriptsize total \space +\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\normalsize (-0.25)\space +\space \normalsize 4.9
\]
\[\large F\scriptsize spring\space=\space\normalsize 4.65 \space N
\]
|
In this force drawing, when the object is at it's lowest point, contrasting the first diagram, the force of the spring now overrules the force of gravity due to superior positioning.
Over to the left, I have set up Newton's 2nd Law to complement this diagram of the object at it's lowest point. The total force is equal to the mass in kilograms times the acceleration in m/s². This is 1.6 Newtons, so keep this aside.
The force of gravity is the mass in kilograms times the rate of gravitational pull in m/s².
The equation of total force equalling the force of the spring minus the force of gravitational pull can be rewritten as the force of gravity equals the total force plus total gravity so that we can solve for the force of the spring.
Adding the total and gravitational forces gives a spring force of 6.5 Newtons, when the object is at it's lowest point during a single period of oscillation.
|
\[\large F\scriptsize{total}\space=\space\normalsize m\space\times\space a
\]
\[\large F\scriptsize{total}\space=\space \normalsize 0.5\space\times\space 3.2
\]
\[\large F\scriptsize{total}\space=\space\normalsize \space1.6\space N
\]
\[\large F\scriptsize{G}\space=\space\normalsize m\space\times\space g
\]
\[\large F\scriptsize{G}\space=\space\normalsize 0.5\space\times\space 9.8
\]
\[\large F\scriptsize{G}\space=\space\normalsize 4.9\space N
\]
\[\large F\scriptsize total\space=\space\large F\scriptsize spring \space -\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\large F\scriptsize total \space +\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\large F\scriptsize total \space +\space \large F\scriptsize G
\]
\[\large F\scriptsize spring\space=\space\normalsize 1.6\space +\space \normalsize 4.9
\]
\[\large F\scriptsize spring\space=\space\normalsize 6.5 \space N
\]
|
Force vs. Time Graph (one Period)
This graph of Force vs. Time looks familiar, because the earlier Acceleration vs. Time Graph is interconnected with this, as Force = Mass * Acceleration. The object's movement can be traced on this graph too, despite the rocky edges of Tracker's inability to catch certain frames. As it goes down, it speeds up, and for a brief second, there is none, but it accelerates once it begins to reascend. When the force is at it's greatest amount, the spring is at the bottom of it's bounce, as gravity acts the most at this point. When at it's minimum value, the object will be found at the top of it's spring, and thus with the lowest force. At equilibrium, the mass is in the middle. |