A particle is executing a simple harmonic motion. Its maximum acceleration is and maximum velocity is .
Then its time period of vibration will be:
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As, we know, in SHM
Maximum acceleration of the particle,
Maximum velocity,
A particle executes linear simple harmonic motion with an amplitude of . When the particle is at from the mean position, the magnitude of its velocity is equal to that of its acceleration.
Then its time period in seconds is
Given, Amplitude
When particle is at
According to question, magnitude of velocity acceleration
A spring of force constant is cut into lengths of ratio . They are connected in series and the new force constant is . Then they are connected in parallel and force constant is . Then is
Let be the complete length of the spring.
Length when cut in ratio, are and
Spring constant
Spring constant for given segments
and
When they are connected in series
And when they are connected in parallel
Then the ratios
A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is at a distance of from the mean position. The time period of oscillation is
From question, acceleration, , and
displacement,
Time period of pendulum,
Average velocity of a particle executing SHM in one complete vibration is:
zero
Displacement of the particle in one complete vibration is zero, so, average velocity in one complete vibration will be
The displacement of a particle executing simple harmonic motion is given by .
Then the amplitude of its oscillation is given by :
.
Given equation
Now assume
which is S.H.M.
where
so, resultant amplitude
The phase difference between displacement and acceleration of a particle in a simple harmonic motion is :
Displacement equation of a SHM
Velocity,
Acceleration,
or,
Hence, phase difference between displacement and acceleration is .
A body is executing simple harmonic motion with frequency ' ', the frequency of its potential energy is
Displacement of particle executing
SHM
Potential energy
From eq. (i) and (ii),
Time period of
Frequency
and time period of
Note : Potential energy varies periodically with double the frequency of SHM. It is maximum and equal to total
A spring is stretched by by a force . The time period of the oscillations when a mass of is suspended by it is
And for spring-mass system undergoing SHM
given
Two pendulums of length and start vibrating in phase. At some instant, the two are at their mean position in the same phase. The minimum number of vibrations of the shorter pendulum after which the two are again in phase at the mean position is:
9
10
8
11
Hence, smaller pendulum will make 1 oscillation morethan longer pendulum.
If longer makes oscillation, then smaller makes oscillation.For oscillating in same phase
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