Engineering Mechanics

Chapter 26 : Helical Springs and Pendulums „„„„„ 535

Let P 1 = Weight shared by the upper spring, and

P 2 = Weight shared by the lower spring.

∴ Elongation of the upper spring,

1

1

1

P

C

δ= ...(i)

and shortening of the lower spring,

2

2

2

P

C

δ= ...(ii)

Since elongation of the upper spring is equal to shortening of the lower spring, therefore
equating the values of δ 1 and δ 2 from the above equations.

12

12

PP

CC

=

∴ P 1 C 2 = P 2 C 1

(P – P 2 ) C 2 = P 2 C 1 ...(Q P 1 + P 2 = P)

P C 2 – P 2 C 2 = P 2 C 1

P 2 (C 1 + C 2 ) = P C 2

or

2

212

P P

CCC

=

+

∴ Displacement

12

P

CC

=

+

2

2

2

...

P

C

⎛⎞

⎜⎟=δ

⎝⎠

Q

We know that the period of vibration,

12

Displacement

22

Acceleration ( )

P

gC C

=π =π

+

Ans.

26.4. SIMPLE PENDULUM

A simple pendulum, in its simplest form, consists of a heavy bob suspended at the end of a
light inextensible, flexible string and the other end of the string
is fixed at O as shown in Fig 26.4.

Let l= Length of the string in metres,
and
m= Mass of the bob in kg (such
that its weight is m.g newtons).
We know that the pendulum is in equilibrium, when
the bob is at A. If the bob is brought to B or C and released, it
will start vibrating between the positions B and C with A as
the mean position.

It has been observed that if the angle* ∠ AOC is very
small, the bob will move with simple harmonic motion. Fig. 26.5.

* If this angle is less than 4°.