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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 cetnre 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...
Given information:
- Length of the rod = 3 m
- Mass per unit length is directly proportional to the distance x from one end of the rod
To find:
The center of gravity of the rod from that end
Solution:
1. Understanding the problem:
The problem states that the mass per unit length of the rod is directly proportional to the distance x from its one end. This means that as we move away from one end of the rod, the mass per unit length increases proportionally.
2. Finding the center of gravity:
The center of gravity of an object is the point at which the weight of the object can be considered to act. In this case, the weight of the rod is distributed along its length, with different sections having different masses.
To find the center of gravity of the rod, we need to consider how the mass is distributed along its length. Since the mass per unit length is directly proportional to the distance from one end, we can assume that the mass at any point x along the rod is given by m(x) = kx, where k is the constant of proportionality.
3. Calculating the constant of proportionality:
To find the constant of proportionality, we can use the given information that the length of the rod is 3 m. Since the mass per unit length is directly proportional to the distance x, we can set up the following equation:
m(x) = kx
We know that the total mass of the rod is given by:
m_total = ∫(0 to 3) m(x) dx
Substituting m(x) = kx, we get:
m_total = ∫(0 to 3) kx dx
m_total = k * ∫(0 to 3) x dx
m_total = k * [x^2/2] (0 to 3)
m_total = k * [(3^2/2) - (0^2/2)]
m_total = k * (9/2)
Since the total mass of the rod is constant, we can set it equal to the known length of the rod:
m_total = 3 * k = 9/2
Solving for k, we get:
k = (9/2) / 3
k = 3/2
4. Calculating the center of gravity:
Now that we have the value of k, we can calculate the center of gravity of the rod. The center of gravity is given by the equation:
x̄ = ∫(0 to 3) x * m(x) dx / m_total
Substituting m(x) = kx and m_total = 3 * k, we get:
x̄ = ∫(0 to 3) x * kx dx / (3 * k)
x̄ = k/3 * ∫(0 to 3) x^2 dx
x̄ = k/3 * [x^3/3] (0 to 3)
x̄ = k/3 * [(3^3/3) - (0^3/3)]
x̄ = k/3 * (27/3)
x̄ = (3/2)/3 * (27/3)
x
- Length of the rod = 3 m
- Mass per unit length is directly proportional to the distance x from one end of the rod
To find:
The center of gravity of the rod from that end
Solution:
1. Understanding the problem:
The problem states that the mass per unit length of the rod is directly proportional to the distance x from its one end. This means that as we move away from one end of the rod, the mass per unit length increases proportionally.
2. Finding the center of gravity:
The center of gravity of an object is the point at which the weight of the object can be considered to act. In this case, the weight of the rod is distributed along its length, with different sections having different masses.
To find the center of gravity of the rod, we need to consider how the mass is distributed along its length. Since the mass per unit length is directly proportional to the distance from one end, we can assume that the mass at any point x along the rod is given by m(x) = kx, where k is the constant of proportionality.
3. Calculating the constant of proportionality:
To find the constant of proportionality, we can use the given information that the length of the rod is 3 m. Since the mass per unit length is directly proportional to the distance x, we can set up the following equation:
m(x) = kx
We know that the total mass of the rod is given by:
m_total = ∫(0 to 3) m(x) dx
Substituting m(x) = kx, we get:
m_total = ∫(0 to 3) kx dx
m_total = k * ∫(0 to 3) x dx
m_total = k * [x^2/2] (0 to 3)
m_total = k * [(3^2/2) - (0^2/2)]
m_total = k * (9/2)
Since the total mass of the rod is constant, we can set it equal to the known length of the rod:
m_total = 3 * k = 9/2
Solving for k, we get:
k = (9/2) / 3
k = 3/2
4. Calculating the center of gravity:
Now that we have the value of k, we can calculate the center of gravity of the rod. The center of gravity is given by the equation:
x̄ = ∫(0 to 3) x * m(x) dx / m_total
Substituting m(x) = kx and m_total = 3 * k, we get:
x̄ = ∫(0 to 3) x * kx dx / (3 * k)
x̄ = k/3 * ∫(0 to 3) x^2 dx
x̄ = k/3 * [x^3/3] (0 to 3)
x̄ = k/3 * [(3^3/3) - (0^3/3)]
x̄ = k/3 * (27/3)
x̄ = (3/2)/3 * (27/3)
x
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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 cetnre of gravity of the rod from that end will be ata)1.5 mb)2 mc)2.5 md)3.0 mCorrect answer is option 'B'. Can you explain this answer?
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