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A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.?
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A rod of length L is placed along the x axis between x=0 and x=L. THE ...
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A rod of length L is placed along the x axis between x=0 and x=L. THE ...
Linear Density and Units
The linear density of the rod is given by the equation:
rho = a + bx,
where 'x' is the distance along the rod.
To determine the SI units of 'a' and 'b':
- Linear Density (rho): The SI unit of linear density is kg/m.
- Coefficient 'a': Since 'rho' must have the dimensions of kg/m, 'a' must also have units of kg/m.
- Coefficient 'b': The term 'bx' must also yield units of kg/m. Therefore, as 'x' is in meters (m), 'b' must have units of kg/m² to ensure that 'b*x' results in kg/m.
Thus, we conclude that:
- a: kg/m
- b: kg/m²
Mass of the Rod
To find the mass of the rod, we need to integrate the linear density over the length of the rod from x=0 to x=L.
- Mass (M) is given by the integral of linear density over the length:
M = ∫(from 0 to L) rho dx
Substituting the expression for rho:
M = ∫(from 0 to L) (a + bx) dx
- Calculating the Integral:
- The integral of 'a' over dx is: a * L
- The integral of 'bx' over dx is: (b/2) * L²
Putting it together:
M = aL + (b/2)L²
Thus, the mass of the rod in terms of 'a', 'b', and 'L' is:
M = aL + (b/2)L².
The linear density of the rod is given by the equation:
rho = a + bx,
where 'x' is the distance along the rod.
To determine the SI units of 'a' and 'b':
- Linear Density (rho): The SI unit of linear density is kg/m.
- Coefficient 'a': Since 'rho' must have the dimensions of kg/m, 'a' must also have units of kg/m.
- Coefficient 'b': The term 'bx' must also yield units of kg/m. Therefore, as 'x' is in meters (m), 'b' must have units of kg/m² to ensure that 'b*x' results in kg/m.
Thus, we conclude that:
- a: kg/m
- b: kg/m²
Mass of the Rod
To find the mass of the rod, we need to integrate the linear density over the length of the rod from x=0 to x=L.
- Mass (M) is given by the integral of linear density over the length:
M = ∫(from 0 to L) rho dx
Substituting the expression for rho:
M = ∫(from 0 to L) (a + bx) dx
- Calculating the Integral:
- The integral of 'a' over dx is: a * L
- The integral of 'bx' over dx is: (b/2) * L²
Putting it together:
M = aL + (b/2)L²
Thus, the mass of the rod in terms of 'a', 'b', and 'L' is:
M = aL + (b/2)L².
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A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.?
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A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.?.
A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.?.
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A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.?, a detailed solution for A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.? has been provided alongside types of A rod of length L is placed along the x axis between x=0 and x=L. THE linear density rho of the rod varies with the distance x from the origin as rho=a bx. Find the si units of a and b and find the mass of the road in terms of a,b,L.? theory, EduRev gives you an
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