
A solid rectangular door of uniform thickness is 2 m wide and has a mass of 60 kg. it is hinged about a vertical axis along one of its longer edges. Assume the thickness of the door is small. If the moment of inertia about its hinges of the door is 10n kg $m^2$, find n.
Answer
154.8k+ views
Hint: We will calculate the moment of inertia of the given solid rectangular door of mass M about the centre of mass of door using the formula ${I_{cm}} = \dfrac{{M{b^2}}}{{12}}$ and then we will use the parallel axis theorem to calculate the moment of inertia about the hinges of the door by using the formula: $I = {I_{cm}} + M{d^2}$ where I is the total moment of inertia and Icm is the moment of inertia about the centre of mass and d is the distance between the vertical axis and the centre of mass. After that, we will compare the given value to the obtained value to find the value of n.
Complete step by step solution:
We are given that a solid rectangular door is of uniform thickness.
Its width is given as 2 m.
The mass of the door is also provided which is 60 kg.
The distance between the centre of mass and the vertical axis of the door will be 1 m as we know that the width of the door is 2 m and hence the distance will be halved.
Now, the moment of inertia about the centre of the mass of the rectangular door will be given as: ${I_{cm}} = \dfrac{{M{b^2}}}{{12}}$
We have M = 60 kg and b = 2 m. on putting these values in the above equation, we get
$ \Rightarrow {I_{cm}} = \dfrac{{60 \times {2^2}}}{{12}} = \dfrac{{240}}{{12}} = 20kg{m^2}$
We have calculated the value of Icm and now we will calculate the value of the moment of inertia about the hinges of the door.
We have $I = {I_{cm}} + M{d^2}$, on putting values in this equation, we get
$ \Rightarrow I = 20 + 60 \times {1^2} = 20 + 60 = 80$
Therefore, the moment of inertia about the hinges of the door I = 80 kg$m^2$.
Comparing it with the given value of moment of inertia which is 10n kg$m^2$, we get n = 8.
Note: You may get confused while reading the given problem because both the moment of inertia is given i.e., the moment of inertia about the centre of mass and the moment of inertia about the vertical axis at a certain distance from the centre of mass of the body.
Complete step by step solution:
We are given that a solid rectangular door is of uniform thickness.
Its width is given as 2 m.
The mass of the door is also provided which is 60 kg.
The distance between the centre of mass and the vertical axis of the door will be 1 m as we know that the width of the door is 2 m and hence the distance will be halved.
Now, the moment of inertia about the centre of the mass of the rectangular door will be given as: ${I_{cm}} = \dfrac{{M{b^2}}}{{12}}$
We have M = 60 kg and b = 2 m. on putting these values in the above equation, we get
$ \Rightarrow {I_{cm}} = \dfrac{{60 \times {2^2}}}{{12}} = \dfrac{{240}}{{12}} = 20kg{m^2}$
We have calculated the value of Icm and now we will calculate the value of the moment of inertia about the hinges of the door.
We have $I = {I_{cm}} + M{d^2}$, on putting values in this equation, we get
$ \Rightarrow I = 20 + 60 \times {1^2} = 20 + 60 = 80$
Therefore, the moment of inertia about the hinges of the door I = 80 kg$m^2$.
Comparing it with the given value of moment of inertia which is 10n kg$m^2$, we get n = 8.
Note: You may get confused while reading the given problem because both the moment of inertia is given i.e., the moment of inertia about the centre of mass and the moment of inertia about the vertical axis at a certain distance from the centre of mass of the body.
Recently Updated Pages
JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Chemical Properties of Hydrogen - Important Concepts for JEE Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
If the unit of power is 1Kilo Watt the length is 100m class 11 physics JEE_Main

Which of the following statements is correct if the class 11 physics JEE_Main

The quantity of heat required to heat one mole of a class 11 physics JEE_Main

IIIT JEE Main Cutoff 2024

Photoelectric Effect and Stopping Potential with Work Function and Derivation for JEE

Newton’s Laws of Motion: Three Laws of Motion Explanation with Examples

Other Pages
JEE Advanced 2025 Revision Notes for Mechanics

Ideal and Non-Ideal Solutions Raoult's Law - JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics

A boy wants to throw a ball from a point A so as to class 11 physics JEE_Main

List of Fastest Century In IPL - Cricket League and FAQs

NEET 2025: All Major Changes in Application Process, Pattern and More
