Running head: MOTION OF A SPRIG MOTION 1
MOTION OF A SPRIG MOTION 2
Motion of a Sprig Motion
Name
Institutional Affiliation
Motion of a Sprig Motion
The motion made by springs when loaded agrees with the systems of harmonic motion. The motion is sinusoidal in nature with regions of acceleration and deceleration. The distance through which a spring stretches is primarily governed by the relationship between the force acting on the spring and the spring constant, a concept defined by the hooks law (Thomas, 2002). The simple harmonic motion of the spring has been applied in numerous situations including design of safety systems to prevent sudden loading, electrical implements, and car (Triana and Fajardo, 2013). Research in the working principles of a spring reveals that a spring works within the Newton’s laws of motion and Hooks law. Un unloaded spring can be seen as a body at rest, application of load to the spring is the action, the spring will tend to retain its original shape and size by applying a restoring force, the reaction. The relationship between the magnitude of load and the possible restoring force applied by spring can be related by the distance through the spring moves. This paper intend to study the motion of a spring by considering various laws and equations relating to simple harmonic motion, hooks law and newton laws of motion. Spring will be analyzed in either vertical or horizontal position and ideal conditions assumed.
Hooks Law
According to the hooks law, spring stretches by a value equivalent to the magnitude of the force applied. However, the net distance travelled by a spring a function of the spring constant. By arithmetic regression, the graph relating force and distance traveled is a straight line passing through the center. Hooks law equation of and ideal spring subjected to a stretching force in ideal conditions is defined by the formula.
F= -k x (Thomas, 2002)
Where x is the distance, through which the spring moves while K is the spring constant. K is depends individual springs and can be obtained by calculation or experimentally for instance, stiff spring have higher K thus moves through shorter distance when loaded that spring with smaller K. The negative sign shows that the spring reacts to the pulling force in a direction opposite the direction of the pulling force (Thomas, 2002). For metallic spring, a point at which the distance through which the spring travels when loaded cease to be linearly related to the applied load is called the elastic limit. At this point, the spring suffers permanent mechanical damage and Hooks law does not apply, the spring does not also regain its original dimensions when the load is removed.
Restoring force on a spring
Newton 1st law postulates that a body will continues to move in a straight line in the direction of the applied force unless acted upon by another force (Hall, 2006). This law implies that when a spring is vibrating, it experience forces in both directions of alternating magnitudes allowing it to change directions. For a body moving from left to right, its experiences a rightward force which decreases at it approaches the neutral axis, similarly, a body moving from right to left experiences a leftward force. The body therefore accelerates as it moves towards the neutral axis, decelerated as it moves away from the neutral axis, and finally stops to change direction. As the body approached the neutral axis, the net force acting on the object becomes zero, as the two forces acting on it are equal and opposite; however, its mass allows it to move to the opposite side an imbalance between the forces resume. The motion of a vibrating body can be represented by a sinusoidal time-force graph with pints of zero force.
The sinusoidal motion of the spring
As explained above, the mass hanging on the spring moves is a sinusoidal manner with phases of acceleration, zero acting force, and decoration. While the object at the center line, it experience no restoring force, however, its velocity is maximum and is at maximum momentum forcing it’s apply a stretching force in the direction of its travel (Triana and Fajardo, 2013). The spring cannot maintain vibrating infinitely, as the mass moves from side to side it experience resistance from the air around it thus losses momentum with each stroke (Triana and Fajardo, 2013). The loss of momentum due to friction decapitating energy from the system will lead an eventual stop at the neutral axis (Triana and Fajardo, 2013). Below are graphs showing the relationship between position of the object and time and velocity versus time. The tables below shows that velocity of the spring varies with are amplitude from the neutral axis.
Time
Position
0
0
10
0.173648
20
0.34202
30
0.5
40
0.642788
50
0.766044
60
0.866025
70
0.939693
80
0.984808
90
1
100
0.984808
110
0.939693
120
0.866025
130
0.766044
140
0.642788
150
0.5
160
0.34202
170
0.173648
180
1.23E-16
190
-0.17365
200
-0.34202
210
-0.5
220
-0.64279
230
-0.76604
240
-0.86603
250
-0.93969
260
-0.98481
270
-1
280
-0.98481
290
-0.93969
300
-0.86603
310
-0.76604
320
-0.64279
330
-0.5
340
-0.34202
350
-0.17365
Time
Velocity
0
0
0.174533
0.173648
0.349066
0.34202
0.523599
0.5
0.698132
0.642788
0.872665
0.766044
1.047198
0.866025
1.22173
0.939693
1.396263
0.984808
1.570796
1
1.745329
0.984808
1.919862
0.939693
2.094395
0.866025
2.268928
0.766044
2.443461
0.642788
2.617994
0.5
2.792527
0.34202
2.96706
0.173648
3.141593
1.23E-16
3.316126
-0.17365
3.490659
-0.34202
3.665191
-0.5
3.839724
-0.64279
4.014257
-0.76604
4.18879
-0.86603
4.363323
-0.93969
4.537856
-0.98481
4.712389
-1
4.886922
-0.98481
5.061455
-0.93969
5.235988
-0.86603
5.410521
-0.76604
5.585054
-0.64279
5.759587
-0.5
5.934119
-0.34202
6.108652
-0.17365
Mathematical expression of the Simple Harmonic motion of a spring
From the hooks law and newton first law of motion, the harmonic oscillation motion of a spring can be derived. The basic formula from the hooks law, f= -k x, can be integrated by the newton’s second law of motion. Newton’s second law substitutes the force of applied by its mass and acceleration; it postulates that the momentum of a body is the product of its mass and acceleration (Verlinde, 2011). Thus
ma= – k x
The equation above give a direct relationship between the mass attached to the spring and its acceleration within regard to its position the sinusoidal graph. The equation can be solved by the application of calculus. To solve the problem, derivative of the equation will be used which give:
m (d2x/dx2) = -k x
By cross multiplication (d2x/dx2) + k x/m = 0
Mathematically, the second derivative of the function with respect to x multiplied by the function equal zero as shown in the equation above . The resultant equation therefore obeys the law of calculus. A graph produced by equation of this nature is sinusoidal as depicted by the harmonic motion graphs of the spring, the cosine graph. Thus, the equation can be represented by:
X= a cos (b t) where a and b are constants of the graph. By differentiation;
d x / d t = -a b sin (b t) and the second derivative is given by d2x / dx2 = a b2 cos (b t)
Substituting the elements into the original equation=:
-a b2 cos (b t) + k / m a cos (b t) = 0
Since b 2 = k / m then the conditions for harmonic equations is fulfilled and can be expressed as
X = a cos (? k / m * t)
From the equation of harmonic motion, the motion of the spring can be predicted. The value of x is at maximum value when cosine = 1 or when x=a (Triana and Fajardo, 2013). This implies that a is the amplitude of the spring motion, or the maximum distance covered on either sides of a vibrating spring. From the cosine, the period can be determined by equating x = a whereby t=0. Since this is a cosine graph
T = 2 ? ? (m / k)
And v = 1 / t = 1 / 2? ? k / m
It should be noted that the value of t and the frequency are only dependent on the mass of the object and spring constant defined by hooks law. This mean that a bigger object will oscillate at the same frequency as a smaller object, only the velocity will change when the same spring is used in both cases. The distance covered by the oscillating mass and the speeds at which it moves at different point in the sinusoid represents the motion of the spring. It frequency also resents the intervals between peak points in the cycle.
Conclusion
From the analysis above, the motion of a spring is seen to be directly proportional to load applied. However, the response a spring offers to a load depends on the spring constant. The distance through which a spring can move therefore depends on the stiffness of the material from which the spring is made. Hooks law is seen to be the main determinant of how much a spring can move when loaded. Governed by newton’s laws motion, a spring set into vibration will continue to vibrate unless acted upon by a resisting force. The forces acting on the spring also obeys the first and second law of motion. A force pulling the spring in one direction is resisted by another, produced due to tension on the spring, in the opposite direction. As the mass stretches away, the magnitude of resisting force also increases and finally equalizes the action force. Since the applied is reduced to zero at the highest point of the sinusoidal motion, the spring’s restoring force pulls the load in the opposite direction and reaches zero, however, the load can compress the spring due to momentum gained during the pulling. From the sinusoidal graphs shown above, a spring is observed to be moving in opposite directions at regular intervals with the same magnitude in ideal situation. The total distance covered by a spring is equal to the summation of the displacement in opposite directions.
References
Hall, L. (2006). The laws of motion. New York: Rosen Pub. Group.
Thomas, P. (2002). Harmonic motion: a simple demonstration. Physics Education, 37(6), 537-538.
Triana, C., & Fajardo, F. (2013). Experimental study of simple harmonic motion of a spring-mass system as a function of spring diameter. Revista Brasileira De Ensino De FÃsica, 35(4), 1-8.
Verlinde, E. (2011). On the origin of gravity and the laws of Newton. Journal Of High Energy Physics, 2011(4).
Time Versus Position
Position 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 0 0.17364817766693033 0.34202014332566871 0.49999999999999994 0.64278760968653925 0.76604444311897801 0.8660254037844386 0.93969262078590832 0.98480775301220802 1 0.98480775301220802 0.93969262078590843 0.86602540378443871 0.76604444311897801 0.64278760968653947 0.49999999999999994 0.34202014332566888 0.17364817766693028 1.22514845490862E-16 -0.17364817766693047 -0.34202014332566866 -0.50000000000000011 -0.64278760968653925 -0.7660444431189779 -0.86602540378443837 -0.93969262078590843 -0.98480775301220802 -1 -0.98480775301220813 -0.93969262078590832 -0.8660254037844386 -0.76604444311897812 -0.64278760968653958 -0.50000000000000044 -0.3420201433256686 -0.17364817766693039
Time Versus Velocity
Sin 0 0.17453292519943295 0.3490658503988659 0.52359877559829882 0.69813170079773179 0.87266462599716477 1.0471975511965976 1.2217304763960306 1.3962634015954636 1.5707963267948966 1.7453292519943295 1.9198621771937625 2.0943951023931953 2.2689280275926285 2.4434609527920612 2.6179938779914944 2.7925268031909272 2.9670597283903604 3.1415926535897931 3.3161255787892263 3.4906585039886591 3.6651914291880923 3.839724354387525 4.0142572795869578 4.1887902047863905 4.3633231299858242 4.5378560551852569 4.7123889803846897 4.8869219055841224 5.0614548307835561 5.2359877559829888 5.4105206811824216 5.5850536063818543 5.759586531581 2871 5.9341194567807207 6.1086523819801535 0 0.17364817766693033 0.34202014332566871 0.49999999999999994 0.64278760968653925 0.76604444311897801 0.8660254037844386 0.93969262078590832 0.98480775301220802 1 0.98480775301220802 0.93969262078590843 0.86602540378443871 0.76604444311897801 0.64278760968653947 0.49999999999999994 0.34202014332566888 0.17364817766693028 1.22514845490862E-16 -0.17364817766693047 -0.34202014332566866 -0.50000000000000011 -0.64278760968653925 -0.7660444431189779 -0.86602540378443837 -0.93969262078590843 -0.98480775301220802 -1 -0.98480775301220813 -0.93969262078590832 -0.8660254037844386 -0.76604444311897812 -0.64278760968653958 -0.50000000000000044 -0.3420201433256686 -0.17364817766693039
