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Pendulum - Wikipedia

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back ... The period of swing of a simple gravity pendulum depends on its length, the local strength of  ...

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Units and Dimensions - Dimensional Analysis, Formula, Applications

Units and dimensions - Understand Dimensional analysis with Limitations and ... NCERT Solutions For Class 12 Physics · NCERT Solutions For Class 12 ... The units that are used to measure these fundamental quantities are called ... In any correct equation representing the relation between physical quantities, the ...

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Mechanics - Simple harmonic oscillations | Britannica

If x is the displacement of the mass from equilibrium (Figure 2B), the springs exert a ... Equation (10) is called Hooke's law, and the force is called the spring force. ... Consider a mass m held in an equilibrium position by springs, as shown in Figure 2A. ... The whole process, known as simple harmonic motion, repeats itself ...

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A pendulum with period, T, is kept in an elevator. What will happen ...

Suppose the lift accelerates upward with acceleration . ... What is the effect on the time period of a simple pendulum if it is in an elevator and is ... pi * √(L÷g) where L is the length of the string, g is the acceleration due to gravity and pi is a constant. ... and the only net force remaining on the pendulum bob is its own weight.

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Physical quantities that have the same dimensions may be added or subtracted. Comprehensive understanding of dimension analysis allows one to deduce certain relationships between different physical dimensions and to check derivation, accuracy and dimensional homogeneity.

According to the Principle of Homogeneity, the dimensions of each term of a dimensional equation on both sides should be the same. This theory allows us to convert the units from one form to another.

Given that

Time period of oscillations = T

T depends on length (l), mass and acceleration due to gravity

T \propto l^x g^y m^z   

T=k\ l^x\ g^y\ m^z  where, k- constant ....................(1)

Step 1: Dimensional formula for given equation  

LHS side:  T = seconds

Dimensional formula [math] T = [L^0 T^{1} M^0] [/math]

RHS side:  =k\ l^x\ g^y\ m^z  

Dimensional formula [math] = [L^1]^x [L^1 T^{-2}]^y [M]^z [/math]

[math] = [L^x L^y T^{-2y} M^z] [/math]

[math]=[\ L^{x+y}\ T^{-2y}\ M^{\ z}\ ][/math]

LHS = RHS

[math] [L^0 T^{1} M^0] = [L^{x+y} T^{-2y } M^z] [/math]

Step 2: Deriving the equation for time period of oscillations

Equate the dimensions on both side of the dimensional equation  

x + y = 0 ,    

-2y=1\ \Rightarrow y = \frac{-1}{2}  

z = 0

x + \frac{-1}{2} = 0 \Rightarrow x = \frac{1}{2}

Substituting x, y and z value in equation (1)

Time equation T = k l^{\frac{1}{2}} g^\frac{-1}{2} m^0     

T = k \frac{\sqrt{l}}{\sqrt{g}}

T = k \sqrt{\frac{l}{g}}