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Rod of a rod of mass M and length L is placed on a smooth table another particle of the same mass M strikes the rod with velocity V not in a distance perpendicular to a distance x which is less than l/b from its Centre the particle sticks to the rod and w be the angular velocity of the system after the collision . The maximum possible value of impulse by varying X that can be imported on the particle during the collision the particle sticks to the rod?
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Rod of a rod of mass M and length L is placed on a smooth table anothe...
Introduction:
We are given a rod of mass M and length L placed on a smooth table. Another particle of the same mass M strikes the rod with velocity V at a distance x from its centre. The particle sticks to the rod after the collision. We need to find the maximum possible value of impulse that can be imparted on the particle during the collision.
Analysis:
To solve this problem, we can use the principle of conservation of linear momentum and the principle of conservation of angular momentum.
Conservation of Linear Momentum:
Before the collision, the linear momentum of the particle is given by:
P1 = M * V
After the collision, the linear momentum of the system (rod + particle) is given by:
P2 = (M + M) * V'
Since the particle sticks to the rod, the final velocity of the system (V') will be the same as the initial velocity V.
Therefore, we can write:
P2 = 2M * V
According to the principle of conservation of linear momentum, P1 = P2:
M * V = 2M * V'
V' = V/2
Conservation of Angular Momentum:
The angular momentum of the system before the collision is zero, as the particle is approaching the rod along a line perpendicular to the rod's axis.
After the collision, the angular momentum of the system is given by:
L = Iw
Where I is the moment of inertia of the system and w is the angular velocity.
The moment of inertia of the system can be calculated as the sum of the moment of inertia of the rod and the moment of inertia of the particle.
For the rod, the moment of inertia about its centre of mass is given by:
I_rod = (1/12) * M * L^2
For the particle, the moment of inertia about its centre of mass is given by:
I_particle = M * x^2
Therefore, the moment of inertia of the system is:
I = I_rod + I_particle
= (1/12) * M * L^2 + M * x^2
According to the principle of conservation of angular momentum, the initial angular momentum is equal to the final angular momentum:
0 = I * w
0 = [(1/12) * M * L^2 + M * x^2] * w
0 = (1/12) * M * L^2 * w + M * x^2 * w
Maximum Possible Value of Impulse:
We need to find the maximum possible value of impulse that can be imparted on the particle during the collision. The impulse can be calculated as the change in linear momentum of the particle.
The change in linear momentum can be calculated as:
Impulse = P2 - P1
= 2M * V - M * V
= M * V
Therefore, the maximum possible value of impulse is M * V.
Conclusion:
In this problem, we used the principles of conservation of linear momentum and conservation of angular momentum to find the maximum possible value of impulse that can be imparted on the particle during the collision. The impulse is given by M * V, where M is the mass of the particle and V is its initial velocity.
We are given a rod of mass M and length L placed on a smooth table. Another particle of the same mass M strikes the rod with velocity V at a distance x from its centre. The particle sticks to the rod after the collision. We need to find the maximum possible value of impulse that can be imparted on the particle during the collision.
Analysis:
To solve this problem, we can use the principle of conservation of linear momentum and the principle of conservation of angular momentum.
Conservation of Linear Momentum:
Before the collision, the linear momentum of the particle is given by:
P1 = M * V
After the collision, the linear momentum of the system (rod + particle) is given by:
P2 = (M + M) * V'
Since the particle sticks to the rod, the final velocity of the system (V') will be the same as the initial velocity V.
Therefore, we can write:
P2 = 2M * V
According to the principle of conservation of linear momentum, P1 = P2:
M * V = 2M * V'
V' = V/2
Conservation of Angular Momentum:
The angular momentum of the system before the collision is zero, as the particle is approaching the rod along a line perpendicular to the rod's axis.
After the collision, the angular momentum of the system is given by:
L = Iw
Where I is the moment of inertia of the system and w is the angular velocity.
The moment of inertia of the system can be calculated as the sum of the moment of inertia of the rod and the moment of inertia of the particle.
For the rod, the moment of inertia about its centre of mass is given by:
I_rod = (1/12) * M * L^2
For the particle, the moment of inertia about its centre of mass is given by:
I_particle = M * x^2
Therefore, the moment of inertia of the system is:
I = I_rod + I_particle
= (1/12) * M * L^2 + M * x^2
According to the principle of conservation of angular momentum, the initial angular momentum is equal to the final angular momentum:
0 = I * w
0 = [(1/12) * M * L^2 + M * x^2] * w
0 = (1/12) * M * L^2 * w + M * x^2 * w
Maximum Possible Value of Impulse:
We need to find the maximum possible value of impulse that can be imparted on the particle during the collision. The impulse can be calculated as the change in linear momentum of the particle.
The change in linear momentum can be calculated as:
Impulse = P2 - P1
= 2M * V - M * V
= M * V
Therefore, the maximum possible value of impulse is M * V.
Conclusion:
In this problem, we used the principles of conservation of linear momentum and conservation of angular momentum to find the maximum possible value of impulse that can be imparted on the particle during the collision. The impulse is given by M * V, where M is the mass of the particle and V is its initial velocity.
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Rod of a rod of mass M and length L is placed on a smooth table another particle of the same mass M strikes the rod with velocity V not in a distance perpendicular to a distance x which is less than l/b from its Centre the particle sticks to the rod and w be the angular velocity of the system after the collision . The maximum possible value of impulse by varying X that can be imported on the particle during the collision the particle sticks to the rod?
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Rod of a rod of mass M and length L is placed on a smooth table another particle of the same mass M strikes the rod with velocity V not in a distance perpendicular to a distance x which is less than l/b from its Centre the particle sticks to the rod and w be the angular velocity of the system after the collision . The maximum possible value of impulse by varying X that can be imported on the particle during the collision the particle sticks to the rod? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Rod of a rod of mass M and length L is placed on a smooth table another particle of the same mass M strikes the rod with velocity V not in a distance perpendicular to a distance x which is less than l/b from its Centre the particle sticks to the rod and w be the angular velocity of the system after the collision . The maximum possible value of impulse by varying X that can be imported on the particle during the collision the particle sticks to the rod? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Rod of a rod of mass M and length L is placed on a smooth table another particle of the same mass M strikes the rod with velocity V not in a distance perpendicular to a distance x which is less than l/b from its Centre the particle sticks to the rod and w be the angular velocity of the system after the collision . The maximum possible value of impulse by varying X that can be imported on the particle during the collision the particle sticks to the rod?.
Rod of a rod of mass M and length L is placed on a smooth table another particle of the same mass M strikes the rod with velocity V not in a distance perpendicular to a distance x which is less than l/b from its Centre the particle sticks to the rod and w be the angular velocity of the system after the collision . The maximum possible value of impulse by varying X that can be imported on the particle during the collision the particle sticks to the rod? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Rod of a rod of mass M and length L is placed on a smooth table another particle of the same mass M strikes the rod with velocity V not in a distance perpendicular to a distance x which is less than l/b from its Centre the particle sticks to the rod and w be the angular velocity of the system after the collision . The maximum possible value of impulse by varying X that can be imported on the particle during the collision the particle sticks to the rod? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Rod of a rod of mass M and length L is placed on a smooth table another particle of the same mass M strikes the rod with velocity V not in a distance perpendicular to a distance x which is less than l/b from its Centre the particle sticks to the rod and w be the angular velocity of the system after the collision . The maximum possible value of impulse by varying X that can be imported on the particle during the collision the particle sticks to the rod?.
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