Answer
409.8k+ views
Hint: We will first draw the diagram for the question first. Then we will find the velocity of the bucket when it touches the water. We will apply the law of conservation of energy in order to find the height of the swing above the lowest position.
Formula Used:
We will use the following formulae to solve this problem:-
$MgL=\dfrac{1}{2}M{{v}^{2}}$ .
Complete Step by Step Solution:
We have to draw the following diagram to get the answer:-
From the figure above we are going to analyse the problem properly.
We will get the speed of the bucket when it touches the water, this is also known as the lowest point. On applying law of principle of energy we get
$MgL=\dfrac{1}{2}M{{v}^{2}}$
$\Rightarrow {{v}^{2}}=2gL$
$v=\sqrt{2gL}$ ………….. $(i)$
Now, using the principle of conservation of linear momentum for the given case, using $(i)$ we have
$M\sqrt{2gL}=\left( M+m \right)v_{1}^{{}}$
$\Rightarrow {{v}_{1}}=\dfrac{M\sqrt{2gL}}{(M+m)}$ ………….. $(ii)$ (${{v}_{1}}$ is the velocity of bucket and water when the bucket scoops up )
Again applying conservation of energy we have
$\dfrac{1}{2}(M+m)v_{1}^{2}=(M+m)gh$
$\Rightarrow v_{1}^{2}=2gh$
$\Rightarrow {{v}_{1}}=\sqrt{2gh}$ ………….. $(iii)$
Now, equating equations $(ii)$ and $(iii)$ for the velocity ${{v}_{1}}$ we get,
$\dfrac{M\sqrt{2gL}}{\left( M+m \right)}=\sqrt{2gh}$ …………….. $(iv)$
Solving above equation further we get,
$h={{\left( \dfrac{M}{M+m} \right)}^{2}}L$.
Hence the correct answer for the height of the swing above the lowest point is, $h={{\left( \dfrac{M}{M+m} \right)}^{2}}L$ .
Additional Information:
Inextensible cord is a cord which can be pulled without being extended. Principle of conservation of linear momentum states that total momentum before collision and after collision is equal. That is, the total momentum of an isolated system is always constant. Conservation of energy states that energy can neither be created nor be destroyed but can only be transformed from one form into another.
Note:
We should apply the formula with care and without confusion. It is very important to use the principle of conservation of momentum at the correct position during the solution. We should not get confused between the law of conservation of energy and conservation of momentum.
Formula Used:
We will use the following formulae to solve this problem:-
$MgL=\dfrac{1}{2}M{{v}^{2}}$ .
Complete Step by Step Solution:
We have to draw the following diagram to get the answer:-
![seo images](https://www.vedantu.com/question-sets/e97fcd20-6541-4a2d-a642-7288d20b1c913989033234723146093.png)
From the figure above we are going to analyse the problem properly.
We will get the speed of the bucket when it touches the water, this is also known as the lowest point. On applying law of principle of energy we get
$MgL=\dfrac{1}{2}M{{v}^{2}}$
$\Rightarrow {{v}^{2}}=2gL$
$v=\sqrt{2gL}$ ………….. $(i)$
Now, using the principle of conservation of linear momentum for the given case, using $(i)$ we have
$M\sqrt{2gL}=\left( M+m \right)v_{1}^{{}}$
$\Rightarrow {{v}_{1}}=\dfrac{M\sqrt{2gL}}{(M+m)}$ ………….. $(ii)$ (${{v}_{1}}$ is the velocity of bucket and water when the bucket scoops up )
Again applying conservation of energy we have
$\dfrac{1}{2}(M+m)v_{1}^{2}=(M+m)gh$
$\Rightarrow v_{1}^{2}=2gh$
$\Rightarrow {{v}_{1}}=\sqrt{2gh}$ ………….. $(iii)$
Now, equating equations $(ii)$ and $(iii)$ for the velocity ${{v}_{1}}$ we get,
$\dfrac{M\sqrt{2gL}}{\left( M+m \right)}=\sqrt{2gh}$ …………….. $(iv)$
Solving above equation further we get,
$h={{\left( \dfrac{M}{M+m} \right)}^{2}}L$.
Hence the correct answer for the height of the swing above the lowest point is, $h={{\left( \dfrac{M}{M+m} \right)}^{2}}L$ .
Additional Information:
Inextensible cord is a cord which can be pulled without being extended. Principle of conservation of linear momentum states that total momentum before collision and after collision is equal. That is, the total momentum of an isolated system is always constant. Conservation of energy states that energy can neither be created nor be destroyed but can only be transformed from one form into another.
Note:
We should apply the formula with care and without confusion. It is very important to use the principle of conservation of momentum at the correct position during the solution. We should not get confused between the law of conservation of energy and conservation of momentum.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)