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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
...

For more information, see **Pendulum** - Wikipedia

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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 ...

For more information, see **Units** and **Dimensions** - **Dimensional** Analysis, **Formula**, Applications

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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 ...

For more information, see Mechanics - **Simple** harmonic **oscillations** | Britannica

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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**.

For more information, see A **pendulum** with **period**, T, is kept **in** an elevator. What will happen ...

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}}