Answer
Verified
442.5k+ views
Hint: Find the position of the centre of the mass of the rod and find the potential energy about this point. Thereafter find the change in the potential energy when the rod is rotated through an angle \[\theta .\]
Complete step by step answer:It is given that, mass of the rod = \[m\], Length of the rod = \[l\] and Angle = \[\theta \]
We know that the centre of mass of a uniform rod of length \[l\]above the ground is \[\dfrac{l}{2}.\]
Therefore, the potential energy of the rod is given by \[\dfrac{{ - mgl}}{2}.\]
When, the rod is displaced through an angle \[\theta \], the height of the centre of mass from the ground becomes \[\dfrac{1}{2}\sin \theta .\]
So, the final potential energy of the rod becomes \[\dfrac{{ - mgl}}{2}\sin \theta .\]
Change in the potential energy = Final potential energy – Initial potential energy
\[ = \dfrac{{ - mgl}}{2}\sin \theta - \dfrac{{ - mgl}}{2}\] \[ = \dfrac{{mgl}}{2}\left( {1 - \sin \theta } \right)\]
Hence the correct option is (B).
Note:The centre of mass of a rigid body is defined as a point where the entire mass of the body is supposed to be concentrated; the nature of the motion of the body shall remain unaffected, if all the forces acting on the body were applied directly on the centre of mass of the body. For rigid bodies of regular geometrical shapes and having uniform distribution of mass, the centre of mass is at the geometrical centre. In case of a uniform rod of length \[l\], the centre of mass lies at the centre of the rod.
Complete step by step answer:It is given that, mass of the rod = \[m\], Length of the rod = \[l\] and Angle = \[\theta \]
We know that the centre of mass of a uniform rod of length \[l\]above the ground is \[\dfrac{l}{2}.\]
Therefore, the potential energy of the rod is given by \[\dfrac{{ - mgl}}{2}.\]
When, the rod is displaced through an angle \[\theta \], the height of the centre of mass from the ground becomes \[\dfrac{1}{2}\sin \theta .\]
So, the final potential energy of the rod becomes \[\dfrac{{ - mgl}}{2}\sin \theta .\]
Change in the potential energy = Final potential energy – Initial potential energy
\[ = \dfrac{{ - mgl}}{2}\sin \theta - \dfrac{{ - mgl}}{2}\] \[ = \dfrac{{mgl}}{2}\left( {1 - \sin \theta } \right)\]
Hence the correct option is (B).
Note:The centre of mass of a rigid body is defined as a point where the entire mass of the body is supposed to be concentrated; the nature of the motion of the body shall remain unaffected, if all the forces acting on the body were applied directly on the centre of mass of the body. For rigid bodies of regular geometrical shapes and having uniform distribution of mass, the centre of mass is at the geometrical centre. In case of a uniform rod of length \[l\], the centre of mass lies at the centre of the rod.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE