Question

Consider a simple pendulum . The period of oscillation of the simple pendulum depends on its length and acceleration due to gravity. Derive the expression for period of oscillation by method dimensions

Answer 1

Answer:

Let Time period =T

Mass of the bob = m

Acceleration due to gravity = g

Length of string = L

Let T \alpha m ^{a}g ^{b}L ^{c}Tαm

a

g

b

L

c

[T] \alpha [m] ^{a}[g] ^{b}[L] ^{c}[T]α[m]

a

[g]

b

[L]

c

M^{0}L^{0}T^{1}=M^{a}L^{b}T^{-2b}L^{c}M

0

L

0

T

1

=M

a

L

b

T

−2b

L

c

M^{0}L^{0}T^{1}=M^{a}L^{b+c}T^{-2b}M

0

L

0

T

1

=M

a

L

b+c

T

−2b

⇒a=0 ⇒ Time period of oscillation is independent of mass of the bob

-2b=1

⇒b=-\frac{1}{2}

2

1

b+c = 0

-\frac{1}{2}

2

1

+ c =0

c=\frac{1}{2}

2

1

Giving values to a,b and c in first equation

T \alpha m ^{0}g ^{- \frac{1}{2} }L ^{ \frac{1}{2} }Tαm

0

g

−

2

1

L

2

1

T \alpha \sqrt{ \frac{L}{g} }Tα

g

L

The real expression for Time period is

T =2 \pi \sqrt{ \frac{L}{g} }T=2π

g

L

Therefore time period of oscillation depends only on gravity and length of the string.

Not on mass of the bob.

Answer 2

Answer:

Explanation:

Concept:

An ideal pendulum made up of a point mass free to oscillate without resistance and hung by a weightless, inextensible, perfectly flexible thread as opposed to an actual pendulum.

Given:

The basic pendulum's length and gravitational acceleration affect how long it oscillates for.

Find:

Derive the oscillation period expression using the dimensions technique.

Solution:

Assume that the basic pendulum's oscillation period t relies on the $a^{t h}$  power of its length I and the $b^{t h}$ power of the acceleration caused by gravity. Then, $t=k l^{a} g^{b} \quad \cdots(i)$

$[L . H . S]=[t]=[T]=\left[M^{0} L^{0} T\right]$

$\left[\right.$R. H.S $S=\left[k l^{a} g^{b}\right]=[L]^{a}\left[L T^{-2}\right]^{b}=\left[M^{0} L^{a+b} T^{-2 b}\right]$

When we compare the sizes of M, L, and T on the two sides, we obtain,$a+b=0$ and $-2 b=1$ or $b=-1 / 2$

It follows that,$a=-b=1 / 2$

If we change a and b in equation I we get

$t=k \sqrt{\frac{l}{g}}$

Hence,  the expression for period of oscillation by method dimensions is $t=k \sqrt{\frac{l}{g}}$.

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