We
are solving series of problems based on the concept of oscillations.
Oscillation is a kind of motion where the body oscillates about a fixed point
called mean position and all oscillatory motions are periodic. It means
oscillatory motion is repeated at regular intervals of time. We need to
understand that all oscillatory motions are periodic but all periodic motions
are not oscillatory. If oscillatory motion is also satisfying a condition like
displacement is directly proportional to acceleration and acceleration is
always directed to wards the mean position, we call that kind of oscillatory
motion as simple harmonic motion. Simple pendulum is one example that executes
simple harmonic motion when it is sightly disturbed from its mean position. We
have derived equation for displacement, velocity and acceleration for a body in
simple harmonic motion.
Problem
To
a body in a simple harmonic motion, velocity is represented as shown in the
equation below. We need to measure maximum acceleration that the body can get
in the given conditions.
Solution
We
have all ready derived equation for the velocity of the particle in simple
harmonic motion. We need to get the given equation in the terms of the standard
equation and the problem can be solved as shown in the diagram below.
Problem
A
particle starts from mean position to a new position and it is as shown in the
diagram below. Its amplitude and time period is given to us in the problem. We
need to find the displacement where the velocity is half of the maximum
velocity.
Solution
We
know that the particle in simple harmonic motion has maximum velocity at the
mean position. As per the given problem at a given instant, velocity of the
particle is half of that maximum. Taking that into consideration and
substituting the data in the standard format, we can solve the problem as shown
in the diagram below.
Problem
Two
different particles are in simple harmonic motion and their displacements are represented as the given equations of the problem. We
need to find the resultant amplitude of the combination. Problem is as shown
below.
Solution
When
we add to oscillatory motion, we need to get a oscillatory motion. The
resultant amplitude can be found using the vector addition equation and the
solution is as shown in the diagram below.
Problem
A
simple harmonic oscillator starts from extreme position and covers a half the
displacement in a given time. We need to measure the further time it is going
to take to reach the mean position and the problem is as shown in the diagram
below.
Solution
As
the particle is here starting from the mean position, we need to know that it
has some initial phase that is ninety degree. We know that the particle takes
one forth of the time period to reach from extreme to mean position and to
measure the remaining time to cover half amplitude to, we need to subtract from
it as shown in the diagram below.
Problem
Number
of springs are connected in series as shown in the problem to a a given mass
and the system is allowed to oscillate. We need to measure the time period of
oscillation of that system.
Solution
We
know that when the springs are in series, the force acting on all of them is
same and the extension in the spring is different and it depends on the nature
of the spring. Using the common formula for the time period of the system and
further simplify the problem as shown in the diagram below.
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