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.
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:
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),
$\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $ and
$\theta $ differ by less than 1%, so we can use the small angle approximation
$\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \approx \theta .$ The angle
$\theta $ describes the position of the pendulum. Using the small angle approximation gives an approximate solution for small angles,
Because this equation has the same form as the equation for SHM, the solution is easy to find. The angular frequency is
$\omega =\sqrt{\frac{g}{L}}$
and the period is
$T=2\pi \sqrt{\frac{L}{g}}.$
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
$\theta $ is less than about
$15\text{\xb0}.$ 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
$T=2\pi \sqrt{\frac{L}{g}}$ for
g , assuming only that the angle of deflection is less than
$15\text{\xb0}$ .
Solution
Square
$T=2\pi \sqrt{\frac{L}{g}}$ and solve for
g :
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
$\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \approx \theta $ to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about
$0.5\text{\xb0}$ .
vector has both magnitude & direction but scalar has only magnitude
Manash
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ron
how do I find resultant of four forces at a point
Inusah
use the socatoa rule
kingsley
draw force diagram, then work out the direction of force.
Rongfang
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
a system of mechanics based of Newton laws motion this is easy difenation of mean...
Arzoodan
what is the meaning of single term,mechanics?
jyotirmayee
mechanics is the science related to the behavior of physical bodies when some external force is applied to them
Lalita
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
Suleiman
can any one send me the best reference book for physics?
Prema
concept of physics by HC verma, Fundamentals of Physics, university of physics
Bhaskar
tq u.
Prema
these are the best physics books one can fond both theory and applications.
Bhaskar
can any one suggest best book for maths with lot of Tricks?
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what is the water height in barometer?
SUNEELL
13.5*76 cm. because Mercury is 13.5 times dense than Mercury
LOVE
water is 13.5 times dense than the Mercury
LOVE
plz tell me frnds the best reference book for physics along with the names of authors.
Prema
i recomended the reference book for physics from library University of Dublin or library Trinity college
Arzoodan
A little help here...
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mechanics which follows Newtons law
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The definition of axial and polar vector .
Arpita
polar vector which have a starting point or pt. of applications is,force,displacement
jyotirmayee
axial vector represent rotational effect and act along the axis of rotation b
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 " .