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A rod is of length 3 m and its mass per unit length is directly proportional to the distance x from its one end.The ceter of gravity of the rod from that end will be at
- a)1.5 m
- b)2 m
- c)2.5 m
- d)3.0 m
Correct answer is option 'B'. Can you explain this answer?
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A rod is of length 3 m and its mass per unit length is directly propor...
Solution:
Given, length of the rod = 3 m
Let the mass per unit length of the rod at a distance x from one end be λx.
Let the center of gravity of the rod be at a distance d from the same end.
We know that the center of gravity is given by the formula:
d = (1/M) ∫x dm
Where M is the total mass of the rod and dm is the mass element at a distance x from one end.
Let us assume that the total mass of the rod is M kg.
∴ M = ∫0³ λx dx
= λ ∫0³ x dx
= λ [x²/2] from 0 to 3
= (9λ/2)
Now, let us consider the integral in the formula for the center of gravity.
d = (1/M) ∫x dm
= (1/M) ∫0³ x λx dx
= (λ/M) ∫0³ x² dx
= (λ/M) [x³/3] from 0 to 3
= (λ/3M) (3³ - 0³)
= (9λ/27M)
= (λ/M) (3/3)
= (1/3) d
∴ d = (1/2) × 3
= 1.5 m
Hence, the center of gravity of the rod from that end is 1.5 m.
Therefore, option (B) is the correct answer.
Given, length of the rod = 3 m
Let the mass per unit length of the rod at a distance x from one end be λx.
Let the center of gravity of the rod be at a distance d from the same end.
We know that the center of gravity is given by the formula:
d = (1/M) ∫x dm
Where M is the total mass of the rod and dm is the mass element at a distance x from one end.
Let us assume that the total mass of the rod is M kg.
∴ M = ∫0³ λx dx
= λ ∫0³ x dx
= λ [x²/2] from 0 to 3
= (9λ/2)
Now, let us consider the integral in the formula for the center of gravity.
d = (1/M) ∫x dm
= (1/M) ∫0³ x λx dx
= (λ/M) ∫0³ x² dx
= (λ/M) [x³/3] from 0 to 3
= (λ/3M) (3³ - 0³)
= (9λ/27M)
= (λ/M) (3/3)
= (1/3) d
∴ d = (1/2) × 3
= 1.5 m
Hence, the center of gravity of the rod from that end is 1.5 m.
Therefore, option (B) is the correct answer.
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A rod is of length 3 m and its mass per unit length is directly proportional to the distance x from its one end.The ceter of gravity of the rod from that end will be ata)1.5 m b)2 mc) 2.5 m d)3.0 mCorrect answer is option 'B'. Can you explain this answer?
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