Objective

to study simple harmonic motion by using simple pendulum.

to study the relationships between the period, frequency, amplitude and
length of a simple pendulum.

to determine the acceleration of gravity.

Theory

A simple pendulum may be described ideally as a point mass suspended
by a massless string from some point about which it is allowed to swing
back and forth in a place.
If a pendulum is set in motion so that is swings back and forth, its motion
will be periodic.
– period( T ) is defined the time that it takes to make one complete
oscillation is
-The frequency( f ) of the oscillations is the number of oscillations that
occur per unit time and is the inverse of the period, f = 1/T.
Similarly, the period is the inverse of the frequency, T = l/f
– The amplitude (A) is defined as The maximum distance that the mass
is displaced from its equilibrium position.
When a simple pendulum is displaced from its equilibrium position, there
will be a restoring force that moves the pendulum back towards its
equilibrium position
If the restoring force F is opposite and directly proportional to the
displacement x from the equilibrium position, so that it satisfies the
relationship

1. F=-KX / L

then the motion of the pendulum will be simple harmonic motion and its
period can be calculated using the equation for the period of simple
harmonic motion

2. T = 2π √( m ⁄ k )

It can be shown that if the amplitude of the motion is kept small,
Equation (2) will be satisfied and the motion of a simple pendulum will
be simple harmonic motion, and Equation (2) can be used The restoring force depends on the gravitational force and the displacement
of the mass from the equilibrium position. Consider Figure 1 where a mass
m is suspended by a string of length l and is displaced from its equilibrium
position by an angle θ and a distance x along the arc through which the
mass moves. The gravitational force can be resolved into two components,
one along the radial direction, away from the point of suspension, and one
along the arc in the direction that the mass moves. The component of the
gravitational force along the arc provides the restoring force F and is given
by

3. F = −m g sin θ

where g is the acceleration of gravity
θ is the angle the pendulum is displaced
the minus sign indicates that the force is opposite to the displacement. For
small amplitudes where θ issmall, sinθ can be approximated by θ measured
in radians so that Equation (3) can be written as

4. F = − m gθ

The angle θ in radians is 𝑋 ⁄𝐿, the arc length divided by the length of the
pendulum or the radius of the circle in which the mass moves. The restoring
force is then given by

5. F=mg*(x/L)

and is directly proportional to the displacement x and is in the form of
Equation (1) where 𝐾 = −𝑚𝑔/𝐿
. Substituting this value of k into Equation
(2), the period of a simple pendulum can be found by

6. T = 2π √(L/g)

Apparatus

• A small bob
• light string
• support stand
• stop watch
• meter stick
• vernier caliper

Procedure

1. Use the vernier caliper to measure the diameter of the bob d and from
   this calculate its radius r.
2. Measure the length of the string from the point of suspension p to the
   top of the bob using a meter stick then find the effective length of the
   pendulum L and write it down.
3. Give a small displacement to the pendulum bob to one side from its
   mean position and allow it to swing.
4. Start the stop watch and measure the time for 10 complete oscillations.
5. Change the length of the pendulum and record time for 10 complete
   oscillations.
6. Repeat the previous step several times.
7. Plot a graph between T2(y axis) and L (x axis).
8. Calculate the acceleration of gravity from the slope of the straight line
   where:

   g = 4π² ⁄ Slope