Question

Three simple harmonic motions of equal amplitudes A
and equal time periods in the same direction combine.
The phase of the second motion is 60° ahead of the first
and the phase of the third motion is 60° ahead of the
second. Find the amplitude of the resultant motion.​

Answer 1

QUESTION

Three simple harmonic motions of equal amplitude A and equal time periods in the same direction combine. The phase of the second motion is 60° ahead of the first and the phase of the third motion is 60° ahead of the second. Find the amplitude of the resultant motion.

SOLUTION

It is given that three S.H.M.s of equal amplitudes A and equal time periods are combined in the same direction.

Let

Y

1

,

Y

2

and

Y

3

Y1,Y2 and Y3

be the three vectors representing the motions, as shown in the figure given below.

According to the question:

Angle between

Y

1

and

Y

2

=

60

∘

Angle between Y1 and Y2=60∘

Angle between

Y

2

and

Y

3

=

60

∘

Angle between Y2 and Y3=60∘

By using the vector method, we can find the resultant vector.

Resultant amplitude = Vector sum of the three vectors

= A + A cos 60° + A cos 60°

=

A

+

A

2

+

A

2

=

2

A

=A+A2+A2=2A