Question

How can you consider a SHM with a reference circle ?

• Explain with diagrams
• Provide necessary equations

#Revision Q17​

Answer 1

How can you consider a SHM with a reference circle ?

When a body moves in a path described by a reference circle, it performs SHM. SHM $\longrightarrow$ to and fro movement because of restoring force (proportional to displacement) about a point . Circular Motion (Uniform) $\longrightarrow$ motion tracing the locus of circle about a point with constant speed . So, a 1-D projection (along circle's diameter) of uniform circular motion is a SHM. Here the constant amplitude of SHM is equal to radius of circle.

Refer To Attachment for Diagram.

Here, P executes circular motion where radius $\longrightarrow$ $\rm{a}$ and $\rm{\: \theta=\omega t}$

Where Displacement (angular) is $\rm{\theta}$ , constant velocity (angular) is $\rm{\omega}$ and time period is $\rm{t}$ . Then displacement of projection of P along Y axis $\longrightarrow$ $\rm{y=O P^{\prime}}$

In Triangle $\rm{OP P^{\prime}}$, $\rm{\sin \theta=\frac{PP^{\prime}}{a}}\implies\rm{\sin \theta=\frac{OP^{\prime}}{a}}\implies\rm{\sin \theta=\frac{y}{a}}\implies\rm{y=a \sin \theta}$

• We already know that $\rm{\: \theta=\omega t}$.

Therefore, $\rm{y=a \sin \omega t}$ which is equation of displacement in SHM

$\rm{\cos \omega t=\sqrt{1-\sin ^{2} \omega t}}$

$\to\rm{\cos \omega t=\sqrt{1-\frac{y^{2}}{a^{2}}}=\frac{1}{a} \sqrt{a^{2}-y^{2}}}$

For velocity, $\rm{v=\frac{d y}{d t}=\frac{d}{d t}(a \sin \omega t)=a \omega \cos \omega t}$

$\boxed{\rm{\Rightarrow v=a \omega \frac{1}{a} \sqrt{a^{2}-y^{2}}=\omega \sqrt{\mathbf{a}^{2}-\mathbf{y}^{2}}}}$

For acceleration, $\rm{\alpha=\frac{d v}{d t}=\frac{d}{d t}(a \omega \cos \omega t)=-a \omega^{2} \sin \omega t}$

$\boxed{\rm{\Rightarrow \alpha=-a \omega^{2} \frac{y}{a}=-\omega^{2} y}}$

$\dashrightarrow\textrm{Both of these are same as SHM equations}$

Answer 2

Answer:

Answer:

i) Liquid Ratio = 0.6 : 1

ii) De-bt Equity Ratio = 0.75 : 1

Explanation:

Solution :

Calculate :

i) Liquid Ratio

ii) De-bt Equity Ratio

★ i) Liquid Ratio :

Liquid Ratio = \sf{\dfrac{Liquid \: Assets}{Current \: Liabilities}}

CurrentLiabilities

LiquidAssets

Current Liabilities = ₹ 1,00,000

Inventory = ₹ 1,00,000

Liquid Assets = Current Assets - Inventory

\longrightarrow⟶ 1,60,000 - 1,00,000

\longrightarrow⟶ 60,000

Liquid Assets = ₹ 60,000

\sf{\longrightarrow{\dfrac{Liquid \: Assets}{Current \: Liabilities}}}⟶

CurrentLiabilities

LiquidAssets

\sf{\longrightarrow \: \dfrac{60000}{100000} = \dfrac{0.6}{1}}⟶

100000

60000

=

1

0.6

Liquid Ratio = 0.6 : 1

★ ii) De-bt Equity Ratio :

De-bt Equity Ratio =

\sf{\longrightarrow{\dfrac{Long \: - \: term \: De.bt }{Shareholders \: Fund}}}⟶

ShareholdersFund

Long−termDe.bt

Debentures = ₹ 3,00,000

Shareholders Fund = ₹ 4,00,000

\sf{\longrightarrow{\dfrac{300000}{400000} = \dfrac{0.75}{1}}}⟶

400000

300000

=

1

0.75

De-bt Equity Ratio = 0.75 : 1

Therefore,

i) Liquid Ratio = 0.6 : 1

ii) De-bt Equity Ratio = 0.75 : 1