<< Chapter < Page Chapter >> Page >
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?


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 ° .


  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 .


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 ° .

Got questions? Get instant answers now!

Questions & Answers

A force F is needed to break a copper wire having radius R. The force needed to break a copper wire of radius 2R will be
Lalit Reply
The difference between vector and scaler quantity
Yakubu Reply
vector has both magnitude & direction but scalar has only magnitude
my marunong ba dto mag prove ng geometry
how do I find resultant of four forces at a point
use the socatoa rule
draw force diagram, then work out the direction of force.
In a closed system of forces... Summation of forces in any direction or plane is zero... Resolve if there is a need to then add forces in a particular plane or direction.. Say the x direction... Equate it tk zero
define moment of inertia
Manoj Reply
what is Euler s theorem
Manoj Reply
what is thermocouple?
Manoj Reply
joining of two wire of different material forming two junctions. If one is hot and another is cold the it will produce emf...
joining of two metal of different materials to form a junction in one is hot & another is cold
define dimensional analysis
Dennis Reply
mathematical derivation?
explain what Newtonian mechanics is.
Elizabeth Reply
a system of mechanics based of Newton laws motion this is easy difenation of mean...
what is the meaning of single term,mechanics?
mechanics is the science related to the behavior of physical bodies when some external force is applied to them
SO ASK What is Newtonian mechanics in physics? Newtonian physics, also calledNewtonian or classical mechanics, is the description of mechanical events—those that involve forces acting on matter—using the laws of motion and gravitation formulated in the late seventeenth century by English physicist
can any one send me the best reference book for physics?
concept of physics by HC verma, Fundamentals of Physics, university of physics
tq u.
these are the best physics books one can fond both theory and applications.
can any one suggest best book for maths with lot of Tricks?
what is the water height in barometer?
13.5*76 cm. because Mercury is 13.5 times dense than Mercury
water is 13.5 times dense than the Mercury
plz tell me frnds the best reference book for physics along with the names of authors.
i recomended the reference book for physics from library University of Dublin or library Trinity college
A little help here... . 1. Newton's laws of Motion, are they applicable to motions of all speeds? 2.state the speeds which are applicable to Newtons laws of Motion
mechanics which follows Newtons law
The definition of axial and polar vector .
polar vector which have a starting point or pt. of applications is,force,displacement
axial vector represent rotational effect and act along the axis of rotation b
explain the rule of free body diagram
Mithu Reply
The polar coordinates of a point are 4π/3 and 5.50m. What are its Cartesian coordinates?
Tiam Reply
application of elasticity
Nangbun Reply
a boy move with a velocity of 5m/s in 4s. What is the distance covered by the boy?
anthony Reply
What is the time required for the sun to reach the earth?
24th hr's, your question is amazing joke 😂
velocity 20 m, s
the sun shines always and the earth rotates so the question should specify a place on earth and that will be 24hrs
good nice work
why 20?.
v =distance/time so make distance the subject from the equation
what is differemce between principles and laws
maaz Reply
how can a 50W light bulb use more energy than a 1000W oven?
Opoku Reply
That depends on how much time we use them
Define vector law of addition
Pawan Reply
It states that, " If two vectors are represented in magnitude and direction by the two sides of a triangle, then their resultant is represented in magnitude and direction by the third side of the triangl " .
thanks yaar
And it's formula
vectors addition is a geometric addition
plank constant is what
chin Reply
plank constant is a phisical constant that the central quantum
links energy of a photon to it's wave length
Practice Key Terms 3

Get the best University physics vol... course in your pocket!

Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?