A 1-kg mass is attached to a spring (k = 4 kg/s^2) and the system is allowed to come to rest. The spring-mass system is attached to a machine that supplies an external driving force f(t)=4 cos(omega*t). The system is started from equilibrium, the mass having no initial displacement nor velocity. Ignore any damping forces. (a) Find the position of the mass as a function of time. (b) Place your answer in the form x(1) = A sin(delta*t) sin (omega_bar*t). Select an omega near the natural frequency of the system to demonstrate the "beating" of the system. Sketch a plot that shows the "beats" and include the envelope of the beating motion in your plot (see Exercise 2)
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.
9. A 1-kg mass is attached to a spring (k = 4 kg/s^2) and the
system is allowed to come to rest. The spring-mass system is attached to a machine that supplies an external driving force f(t)=4 cos(omega*t). The system is started
from equilibrium, the mass having no initial displacement
nor velocity. Ignore any damping forces.
(a) Find the position of the mass as a function of time.
(b) Place your answer in the form x(1) = A sin(delta*t) sin (omega_bar*t).
Select an omega near the natural frequency of the system
to demonstrate the "beating" of the system. Sketch a
plot that shows the "beats" and include the envelope
of the beating motion in your plot (see Exercise 2).
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