(^536) A Textbook of Engineering Mechanics
Consider the equilibrium of the system at C. The weight mg
of the bob can be resolved into two components i.e., mg cos θ and
mg sin θ at right angles to each other.
The component mg cos θ will act along the thread. This will
balance the tension in the string as shown in Fig. 26.5. The other
component mg sin θ, being unbalanced, will give rise to an
acceleration in the direction CA.
Let a = Acceleration caused by the
component.
∴ Force responsible for this acceleration
=mg sin θ ...(i)
We also know that the force
= Mass × Acceleration = m.a ...(ii)
Equating equations (i) and (ii),
ma = mg sin θ
∴ a = g sin θ
Since the angle θ is very small, therefore substituting sin θ = θ in the above equation,
a = g θ ...(iii)
From the geometry of the figure, we know that
Length of the arc
Radius
AC
l
θ= =
∴
AC
ag g
l
=θ=×
or
ACl
ag
= ...(where AC = Displacement of the body)
We know that in a simple harmonic motion, the time period,
Displacement
22
Acceleration
AC
t
a
=π =π^2
l
g
=π
...
ACl
ag
⎛⎞
⎜⎟=
⎝⎠
Q
Notes 1.The motion of the bob from one extremity to the other (i.e. from B to C or C to B) is
known as a beat or swing. It is thus obvious, that one beat =^12 oscillation. Therefore time
period for one beat
l
g
=π
2.A pendulum, which executes one beat per second, is known as a second’s pendulum.
26.5. LAWS OF SIMPLE PENDULUM
The following laws of a simple pendulum are important from the subject point of view :
- Law of isochronism. It states, “The time period (t) of simple pendulum does not depend
on its amplitude of vibrations, and remain the same provided the angular amplitude
(θ) does not exceed 4°.”