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The linear mass density of a rod of length L, varies from one end to the other as 1 +
where x is the
distance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two massless
strings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct?
The linear mass density of a rod of length L, varies from one end to the other as 1 +
where x is the
distance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two massless
strings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct?
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2The linear mass density of a rod of length L, varies from one end to ...
Problem Statement: The linear mass density of a rod of length L, varies from one end to the other as 1 where x is the distance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two massless strings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct? Explain in details.
Solution:
Given, the linear mass density of a rod of length L varies from one end to the other as 1 where x is the distance from one end and is a constant.
Let us consider a small element of length dx at a distance x from the lower end of the rod. The mass of the element is dm = λdx where λ is the linear mass density of the rod.
Let T1 and T2 be the tensions in the strings attached at points A and B respectively.
Vertical forces:
The net vertical force on the rod is zero as it is in equilibrium. Therefore, T1 + T2 = weight of the rod.
Horizontal forces:
The horizontal component of tension at points A and B must be equal as the rod is in equilibrium. Therefore, T1cosθ = T2cosθ.
Moment about point A:
Taking moments about point A, we get:
T2Lcosθ = ∫x=0 to L xdm.g
Substituting the value of dm, we get:
T2Lcosθ = ∫x=0 to L x(λdx).g
T2Lcosθ = λg∫x=0 to L xdx
T2Lcosθ = λg(L^2)/2
Moment about point B:
Taking moments about point B, we get:
T1Lcosθ = ∫x=0 to L (L - x)dm.g
Substituting the value of dm, we get:
T1Lcosθ = ∫x=0 to L (L - x)(λdx).g
T1Lcosθ = λg∫x=0 to L (L - x)dx
T1Lcosθ = λg(L^2)/2
Conclusion:
From the above equations, we can see that T1 = T2 and hence, the tensions in the strings are equal. Therefore, statement (a) is correct.
Also, the tensions in the strings are independent of the linear mass density of the rod. Therefore, statement (b) is also correct.
Hence, both statements (a) and (b) are correct.
Solution:
Given, the linear mass density of a rod of length L varies from one end to the other as 1 where x is the distance from one end and is a constant.
Let us consider a small element of length dx at a distance x from the lower end of the rod. The mass of the element is dm = λdx where λ is the linear mass density of the rod.
Let T1 and T2 be the tensions in the strings attached at points A and B respectively.
Vertical forces:
The net vertical force on the rod is zero as it is in equilibrium. Therefore, T1 + T2 = weight of the rod.
Horizontal forces:
The horizontal component of tension at points A and B must be equal as the rod is in equilibrium. Therefore, T1cosθ = T2cosθ.
Moment about point A:
Taking moments about point A, we get:
T2Lcosθ = ∫x=0 to L xdm.g
Substituting the value of dm, we get:
T2Lcosθ = ∫x=0 to L x(λdx).g
T2Lcosθ = λg∫x=0 to L xdx
T2Lcosθ = λg(L^2)/2
Moment about point B:
Taking moments about point B, we get:
T1Lcosθ = ∫x=0 to L (L - x)dm.g
Substituting the value of dm, we get:
T1Lcosθ = ∫x=0 to L (L - x)(λdx).g
T1Lcosθ = λg∫x=0 to L (L - x)dx
T1Lcosθ = λg(L^2)/2
Conclusion:
From the above equations, we can see that T1 = T2 and hence, the tensions in the strings are equal. Therefore, statement (a) is correct.
Also, the tensions in the strings are independent of the linear mass density of the rod. Therefore, statement (b) is also correct.
Hence, both statements (a) and (b) are correct.
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2The linear mass density of a rod of length L, varies from one end to the other as 1 +where x is thedistance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two masslessstrings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct?
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2The linear mass density of a rod of length L, varies from one end to the other as 1 +where x is thedistance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two masslessstrings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 2The linear mass density of a rod of length L, varies from one end to the other as 1 +where x is thedistance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two masslessstrings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 2The linear mass density of a rod of length L, varies from one end to the other as 1 +where x is thedistance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two masslessstrings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct?.
2The linear mass density of a rod of length L, varies from one end to the other as 1 +where x is thedistance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two masslessstrings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 2The linear mass density of a rod of length L, varies from one end to the other as 1 +where x is thedistance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two masslessstrings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 2The linear mass density of a rod of length L, varies from one end to the other as 1 +where x is thedistance from one end (see figure), and is a constant. The rod is suspended from a ceiling by two masslessstrings having tensions Tl and T2. Then, which of the following statement(s) is(are) correct?.
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