- A mass m is attached to one end of a horizontal spring, while the other end is anchored to a wall. The extension x(t) of the spring from its unstretched length determines the horizontal displacement of the mass.
- When undisturbed, the mass is at rest and the spring is unextended. But if displaced from this equilibrium, the spring exerts a restoring force f(x) = -kx on the mass according to Hooke's law, where k is the spring constant.
- Using Newton's second law, F=ma, this results in the mass undergoing simple harmonic motion described by the equation x(t) = A cos(ωt + Φ), where the frequency
This topic is about Free Oscillation.
Spring-Mass system is an application of Simple Harmonic Motion (SHM).
This topic is Depend on the Ordinary Differential Equation.
Topic of computational methods for mechanical engineering. Information about spring mass system. Mathematical modelling of spring mass system. free mass spring system. Damped vibration. Forced damped system. Free oscillation.
This topic is about Free Oscillation.
Spring-Mass system is an application of Simple Harmonic Motion (SHM).
This topic is Depend on the Ordinary Differential Equation.
Topic of computational methods for mechanical engineering. Information about spring mass system. Mathematical modelling of spring mass system. free mass spring system. Damped vibration. Forced damped system. Free oscillation.
Phyisics explaination on simple harmonic motion for first and second year university students , includes practical and theory about waves and some practical applications
Phyisics explaination on simple harmonic motion for first and second year university students , includes practical and theory about waves and some practical applications
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References include links to illustrative youtube clips and other powerpoints that contributed to this peresentation.
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This presentation was uploaded with the author’s consent.
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2. Consider a compact mass m that slides
over a frictionless horizontal surface.
Suppose that the mass is attached to
one end of a light horizontal spring whose
other end is anchored in an immovable wall.
At time t , let x(t) be the extension of the
spring.
Note: the extension of the spring is the difference between the spring's actual length
and its unstretched length.
x(t) can also be used as a coordinate to determine the instantaneous horizontal
displacement of the mass.
3. The equilibrium state of the system
corresponds to the situation in which
the mass is at rest, and the spring is
unextended.
In this state, zero horizontal force acts
on the mass, and so there is no reason
for it to start to move.
However, if the system is perturbed from its equilibrium state (i.e., if the mass is
displaced sideways, such that the spring becomes extended) then the mass
experiences a horizontal force given by Hooke's law,
f(x)= -kx
4. • f(x)= -kx
• Here, k>0 is the so-called force constant of the spring.
• The negative sign in the preceding expression indicates that f(x) is a restoring
force (i.e., if the displacement is positive then the force is negative, and vice
versa).
• The magnitude of this restoring force is directly proportional to the displacement
of the mass from its equilibrium position. Hooke's law only holds for relatively
small spring extensions.
• Newton's second law of motion leads to the following time evolution equation for
the system,
F = ma = -kx
a= -(k/m)x
5. The mass-spring system will undergo simple
harmonic mothion, with the displacement of the
mass given by the standard equation:
x(t) = A cos (ωt + Φ)
Recall the acceleration of a Simple Harmonic
Oscillator : a(t)= -(ω^2A)cos(ωt+Φ)
We obtain: ω^2=k/m
We can see that the frequency of oscillation depends on the stiffness of the
spring (K) and the mass of the oscillating object. A light mass attached to a stiff
spring has a large frequency and, hence, a small period.