We
are solving series of problems based on the concept of oscillation and simple
harmonic motion. If the acceleration of a body is directly proportional to the
displacement and opposite to it, then the motion is said to be simple harmonic
motion. This can be done by many type of systems and if they are obeying the
above mentioned condition, we can get the time period or frequency and it is
constant all over the oscillatory motion. A loaded spring do oscillate when it
is slightly disturbed. In practical way, any body won’t continue its
oscillatory motion for ever and and due to air resistance, it slowly decreases
and finally comes to the state of the rest. This kind of motion is called
damped oscillatory motion and it that has to be continued, it shall have some
external force support and that kind of motion is called forced oscillation.
Problem
We
need to find the time period of pendulum of infinite length and the problem is
as shown in the diagram below.
Solution
We
need to write the equation for the torque as the product of force and
perpendicular distance. Force is nothing but the weight of the body and
distance is found as shown in the diagram below. We need to write further
torque as the product of moment of inertia and angular acceleration basing on
its definition. Thus we can equation for the acceleration and hence the time
period as shown in the diagram below.
Problem
Frequency
of a particle in SHM is given to us as shown below. We need to find the maximum
speed that the particle can reach in the oscillatory motion.
Solution
The
restoring force acting on the spring is nothing but the weight of the body and
hence we can find the maximum possible displacement by equating them. By
comparing that with the standard equation, we can find angular velocity and
hence the maximum speed of the particle as shown in the diagram below.
Problem
A
particle of known mass is attached to three springs as shown in the diagram
below. If the particle is slightly disturbed, we need to know the time period
of the system.
Solution
The
resultant force acting on the mass using the vector laws of addition. We need
to resolve the force into components and add as shown in the diagram below. By
equating it to the restoring force, we can write the equation for the time
period as shown in the diagram below.
Problem
Two
blocks are kept one over the other and the lower surface is smooth and the
connecting surface is rough as shown in the diagram below. The system is
connected to a rigid support with a spring and the time period of the system is
given to us in the problem. We need to find the mass of the upper block and the
coefficient of friction so that there is no slipping between the two blocks.
Solution
We
know the equation for the time period of a loaded spring and using that data,
we can find the mass of the upper body as shown in the diagram below. Further
equating the frictional force to the restoring force, we can solve the problem
as shown in the diagram below.
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