(b) Shortly discus its kinetic and potential energies at different displacements.
Ans. (a) Motion of a mass attached with horizontal spring mass system
let us consider a mass attached at one end of a spring while other end of spring is fixed with right body. The mass is placed on a horizontal friction less plane, so that its weight and the reaction of plane being equal and opposite is displaced from its mean position by applying a force F and then its is released then mass starts vibratory motion.
At any instant the displacement of mass is denoted by x (which is also equal to change length of spring). By using Hock's Law we write the value of restoring force applied by the spring on the mass attached is.
written as follows.
The applied force F = kx
The restoring force F = -kx
Here, k is called spring constant its value is directly proportional to stiffness of spring and its is also called resorting force per unit change of length of the spring. Its unit is N/m.
The negative sign shows that restoring force due to displacement is always in opposite direction of the displacement. It is explained by figure. We can say that resorting force has direction towards mean position is simple harmonic motion.
Definition of Simple Harmonic Motion:
The vibratory motion of a body for which the force acting on a body is proportional to its displacement from mean position and its always directed towards mean positin is called simple harmonic motion.
During simple harmonic motino the restoring force is zero while passing through mean postion (at x =0 ) and its valuse is mazimum possible when its is at xtreame positino.
During simple harmonic motino we can also prove that acceleration of body is directly proportional to displacement and its directed twoard the mean position. It is done by using Newton's 2nd law of motion (F = ma), in Hock's law we get
-ma = kx
Dividing both sides by (m) we get
a = -(k/m)x where k/m = constant
a = - ( constant ) x
it is the mathematical equation for SHM. It states that acceleration of the body is directly proportional to the displacement and directed to the mean position
The motion of mass attached with spring has time period whose values is given bellow.
T = 2pi (k/m)root
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