The position of a certain spring-mass system satisfies the initial value problem 3/2u''+ku=0, u(0)=2 , u'(0)=v If the period and amplitude of the resulting motion are observed to be pi and 3, respectively, determine the values of k and v.
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 position of a certain spring-mass system satisfies the initial value problem
3/2u''+ku=0, u(0)=2 , u'(0)=v
If the period and amplitude of the resulting motion are observed to be pi and 3, respectively, determine the values of k and v.
Discuss the relation between the spring's constant and the initial velocity with the period and the amplitude.
What does it happen about the period when there is a change in k but keeps the mass m fixed?
What does it happen about the amplitude when there is a change in v?
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