The spring constant Context You do an internship in a mechanical component characterization company for your engineering degree. You are currently working on determining the spring constant of industrial springs. To do this, you measure the elongation of springs placed on a graduated inclined plane, to which you attach masses. Constraints One end of the spring is attached to a hook at the top of the inclined plane and the other end is attached to a mass. The spring is parallel to the plane, and the force it exerts on the mass is also parallel to the plane. The angle of inclination of the plane is known, as well as its uncertainty.
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
The spring constant
Context
You do an internship in a mechanical component characterization company for your engineering degree. You are currently working on determining the spring constant of industrial springs. To do this, you measure the elongation of springs placed on a graduated inclined plane, to which you attach masses.
Constraints
One end of the spring is attached to a hook at the top of the inclined plane and the other end is attached to a mass.
The spring is parallel to the plane, and the force it exerts on the mass is also parallel to the plane.
The angle of inclination of the plane is known, as well as its uncertainty.
There is no friction between the plane and the mass.
The natural length of the spring and its uncertainty are known.
By hooking a known mass (m ± delta m), you measure that the spring now has a length (L ± delta L).
We use the value g=(9.81 ± 0.01)m/s^2 for our calculations.
Schematization
Draw a diagram of the object that interests us. Draw your x and y axes. Draw and name each force experienced by the object that interests us.
Modelization
Build a model to find the spring constant of the spring (with its uncertainty) given the known parameters. Then test your model with the following values:
Plane tilt angle: (11.4 ± 0.2) degrees
Natural spring length: (0.145 ± 0.001) m
Attached mass: (0.15 ± 0.001) kg
Spring length with a mass attached: (0.17 ± 0.002) m
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