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A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone . Both the sphere and the cone are made of the same material ( ). Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg,,g=10m/s2)?
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A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m...
Problem Statement
A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone. Both the sphere and the cone are made of the same material. Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg, g=10m/s2).
Solution
Step 1: Finding the Potential Energy of the System
The potential energy of the system can be found using the formula:
U = mgh
Where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the ground.
Since the arrangement is fixed to the ground at the apex of the cone, the height of the object above the ground can be taken as the height of the apex of the cone above the ground.
Let H be the height of the apex of the cone above the ground. Then, the potential energy of the system can be written as:
U = 2mgh
Where the factor of 2 is added because there are two objects in the system (the sphere and the cone).
Step 2: Finding the Height of the Apex of the Cone
The height of the apex of the cone above the ground can be found using the formula for the height of a cone:
H = 3Rtan(30)
Where R is the radius of the cone and 30 is the semi-vertical angle of the cone. Substituting the given values, we get:
H = 3R(1/√3)
H = R√3
Step 3: Finding the Potential Energy of the System at Room Temperature
The potential energy of the system at room temperature can be found by substituting the values of m, g, and H in the formula for potential energy:
U1 = 2mgh
U1 = 2(5)(10)(R√3)
U1 = 100R√3
Step 4: Finding the Change in Height Due to Temperature Increase
When the temperature of the system is increased, both the sphere and the cone will expand. Let ΔR be the increase in radius of both the sphere and the cone.
Using the formula for linear expansion, the increase in height of the cone can be found as:
ΔH = H(αΔT)
Where α is the coefficient of linear expansion and ΔT is the increase in temperature.
Since the sphere and the cone are made of the same material, they have the same coefficient of linear expansion. Let α be the coefficient of linear expansion of the material.
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A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m...
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A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone . Both the sphere and the cone are made of the same material ( ). Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg,,g=10m/s2)?
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A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone . Both the sphere and the cone are made of the same material ( ). Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg,,g=10m/s2)? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone . Both the sphere and the cone are made of the same material ( ). Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg,,g=10m/s2)? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone . Both the sphere and the cone are made of the same material ( ). Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg,,g=10m/s2)?.
A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone . Both the sphere and the cone are made of the same material ( ). Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg,,g=10m/s2)? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone . Both the sphere and the cone are made of the same material ( ). Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg,,g=10m/s2)? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A hollow sphere (mass m, Radius R) is put inside a hollow cone (mass m, radius 3R and semi-vertical Angle 300) as shown in the figure. The whole arrangement is rigidly fixed to ground at the apex of the cone . Both the sphere and the cone are made of the same material ( ). Considering the major changes in dimensions could happen only due to heating effects, find the increase in potential energy of the system if its temperature is increased by 1000 C (in joule)( m=5 kg,,g=10m/s2)?.
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