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Consider a mass attached to a spring. We know that if the mass is at rest it will remain there. Let the equilibrium position be x = 0. Now if we displace the mass towards right and release, the spring accelerates the mass towards the center position i.e x = 0. At this position the force is zero. But since the mass is moving with a certain velocity, it overshoots to the other side. The spring slows down the movement of the mass and brings it to rest at an extreme left position. Now due to the spring force, the mass accelerates towards the right. Neglecting friction, this to and fro oscillations continue. This is called simple harmonic motion.
From Newton's second law
which is differential equation of second order and first degree.
If 'x' is the displacement and 'A' the amplitude, the SHM is described by sinusoidal function as
If a ball is thrown vertically with a velocity of 30 m/s, find the maximum height the ball reaches. (neglect air resistance and assume g = 10 m/sec2)
Sol: Let 'v' be the velocity and 'h' the height of the ball at any time 't'.
The acceleration is dv/dt and the acceleration due to gravity(in the upward direction) is – g.
∴ dv/dt = – g
or dv = – gdt
The initial velocity v1 = 30 m/s at t1 = 0
(v – 30) = –10(t – 0)
v = 30 – 10t
Substituting v = dh/dt,
dh/dt = 30 – 10t
or dh = (30 – 10t)dt