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By the end of this section, you will be able to:
  • State the forces that act on a simple pendulum
  • Determine the angular frequency, frequency, and period of a simple pendulum in terms of the length of the pendulum and the acceleration due to gravity
  • Define the period for a physical pendulum
  • Define the period for a torsional pendulum

Pendulums are in common usage. Grandfather clocks use a pendulum to keep time and a pendulum can be used to measure the acceleration due to gravity. For small displacements, a pendulum is a simple harmonic oscillator.

The simple pendulum

A simple pendulum    is defined to have a point mass, also known as the pendulum bob , which is suspended from a string of length L with negligible mass ( [link] ). Here, the only forces acting on the bob are the force of gravity (i.e., the weight of the bob) and tension from the string. The mass of the string is assumed to be negligible as compared to the mass of the bob.

In the figure, a horizontal bar is shown. A string of length L extends from the bar at an angle theta counterclockwise from the vertical. The vertical direction is indicated by a dashed line extending down from where the string is attached to the bar. A circular bob of mass m is attached to the lower end of the string. The arc from the mass to the vertical is indicated by another dashed line and is a length s. A red arrow showing the time T of the oscillation of the mob is shown along the string line toward the bar. A coordinate system is shown near the bob with the positive y direction aligned with the string and pointing toward the pivot point and the positive x direction pointing tangent to the arc and away from the equilibrium position. An blue arrow from the bob toward the pivot, along the string, is labeled F sub T. A red arrow from the bob pointing down is labeled w = m g. A red arrow pointing tangent to the arc and toward equilibrium, in the minus x direction, is labeled minus m g sine theta. A red arrow at an angle theta counterclockwise from w is labeled minus m g cosine theta.
A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough not to stretch appreciably. The linear displacement from equilibrium is s , the length of the arc. Also shown are the forces on the bob, which result in a net force of m g sin θ toward the equilibrium position—that is, a restoring force.

Consider the torque on the pendulum. The force providing the restoring torque is the component of the weight of the pendulum bob that acts along the arc length. The torque is the length of the string L times the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement:

τ = L ( m g sin θ ) ; I α = L ( m g sin θ ) ; I d 2 θ d t 2 = L ( m g sin θ ) ; m L 2 d 2 θ d t 2 = L ( m g sin θ ) ; d 2 θ d t 2 = g L sin θ .

The solution to this differential equation involves advanced calculus, and is beyond the scope of this text. But note that for small angles (less than 15 degrees), sin θ and θ differ by less than 1%, so we can use the small angle approximation sin θ θ . The angle θ describes the position of the pendulum. Using the small angle approximation gives an approximate solution for small angles,

d 2 θ d t 2 = g L θ .

Because this equation has the same form as the equation for SHM, the solution is easy to find. The angular frequency is

ω = g L

and the period is

T = 2 π L g .

The period of a simple pendulum depends on its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass and the maximum displacement. As with simple harmonic oscillators, the period T for a pendulum is nearly independent of amplitude, especially if θ is less than about 15 ° . Even simple pendulum clocks can be finely adjusted and remain accurate.

Note the dependence of T on g . If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity, as in the following example.

Measuring acceleration due to gravity by the period of a pendulum

What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?

Strategy

We are asked to find g given the period T and the length L of a pendulum. We can solve T = 2 π L g for g , assuming only that the angle of deflection is less than 15 ° .

Solution

  1. Square T = 2 π L g and solve for g :
    g = 4 π 2 L T 2 .
  2. Substitute known values into the new equation:
    g = 4 π 2 0.75000 m ( 1.7357 s ) 2 .
  3. Calculate to find g :
    g = 9.8281 m/s 2 .

Significance

This method for determining g can be very accurate, which is why length and period are given to five digits in this example. For the precision of the approximation sin θ θ to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about 0.5 ° .

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Questions & Answers

is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
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A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
who is Newton?
John Reply
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
Daniel
and the color spectrum
Daniel
what is a scalar quantity
Peter Reply
scalar: are quantity have numerical value
muslim
is that a better way in defining scalar quantity
Peter
thanks
muslim
quantity that has magnitude but no direction
Peter
upward force and downward force lift
adegboye Reply
upward force and downward force on lift
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hi
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Helo
Tobi
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sununu
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Nessy
1.609km in 1mile
Faqir
what's the si unit of impulse
Iguh Reply
The Newton second (N•s)
Ethan
what is the s. I unit of current
Roland Reply
Amphere(A)
imam
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Roland
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imam
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively
Pallavi Reply
what is head to tail rule?
kinza Reply
Explain Head to tail rule?
kinza
Practice Key Terms 3

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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