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Any distributed mass can be replaced by two point masses to have same dynamical properties if
- a)if sum of the two masses is equal to the total mass
- b)the combined center of mass concides with that of the rod
- c)the moment of inertia of two point masses about the perpendicular axis through their combined center of mass is equal to that of the rod
- d)all of the above
Correct answer is option 'D'. Can you explain this answer?
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Any distributed mass can be replaced by two point masses to have same ...
The answer is option 'D': all of the above.
Explanation:
To understand why option 'D' is correct, let's break down each statement:
a) If the sum of the two masses is equal to the total mass:
When a distributed mass is replaced by two point masses, the sum of the masses of the point masses should be equal to the total mass of the distributed mass. This is because the total mass of the system should remain the same for the dynamical properties to be equivalent.
b) If the combined center of mass coincides with that of the rod:
The center of mass of a distributed mass system is the point where the entire mass can be considered to be concentrated. When replacing the distributed mass with two point masses, the combined center of mass of the two point masses should coincide with the center of mass of the original distributed mass. This ensures that the overall distribution of mass remains the same.
c) If the moment of inertia of two point masses about the perpendicular axis through their combined center of mass is equal to that of the rod:
The moment of inertia is a measure of an object's resistance to rotational motion. When replacing a distributed mass with two point masses, the combined moment of inertia of the two point masses about the perpendicular axis through their combined center of mass should be equal to the moment of inertia of the original distributed mass. This ensures that the rotational behavior of the system remains the same.
Combining all the statements:
If all three conditions are satisfied, it means that the replacement of the distributed mass with two point masses reproduces all the dynamical properties of the original distributed mass. The sum of the masses is equal, the center of mass coincides, and the moment of inertia about the perpendicular axis is equal. Therefore, option 'D' is the correct answer.
In summary, to have the same dynamical properties, a distributed mass can be replaced by two point masses if the sum of the masses is equal to the total mass, the combined center of mass coincides with that of the rod, and the moment of inertia of the two point masses about the perpendicular axis through their combined center of mass is equal to that of the rod.
Explanation:
To understand why option 'D' is correct, let's break down each statement:
a) If the sum of the two masses is equal to the total mass:
When a distributed mass is replaced by two point masses, the sum of the masses of the point masses should be equal to the total mass of the distributed mass. This is because the total mass of the system should remain the same for the dynamical properties to be equivalent.
b) If the combined center of mass coincides with that of the rod:
The center of mass of a distributed mass system is the point where the entire mass can be considered to be concentrated. When replacing the distributed mass with two point masses, the combined center of mass of the two point masses should coincide with the center of mass of the original distributed mass. This ensures that the overall distribution of mass remains the same.
c) If the moment of inertia of two point masses about the perpendicular axis through their combined center of mass is equal to that of the rod:
The moment of inertia is a measure of an object's resistance to rotational motion. When replacing a distributed mass with two point masses, the combined moment of inertia of the two point masses about the perpendicular axis through their combined center of mass should be equal to the moment of inertia of the original distributed mass. This ensures that the rotational behavior of the system remains the same.
Combining all the statements:
If all three conditions are satisfied, it means that the replacement of the distributed mass with two point masses reproduces all the dynamical properties of the original distributed mass. The sum of the masses is equal, the center of mass coincides, and the moment of inertia about the perpendicular axis is equal. Therefore, option 'D' is the correct answer.
In summary, to have the same dynamical properties, a distributed mass can be replaced by two point masses if the sum of the masses is equal to the total mass, the combined center of mass coincides with that of the rod, and the moment of inertia of the two point masses about the perpendicular axis through their combined center of mass is equal to that of the rod.
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Any distributed mass can be replaced by two point masses to have same dynamical properties ifa)if sum of the two masses is equal to the total massb)the combined center of mass concides with that of the rodc)the moment of inertia of two point masses about the perpendicular axis through their combined center of mass is equal to that of the rodd)all of the aboveCorrect answer is option 'D'. Can you explain this answer?
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