Problem

Point masses m,4m, and 2m are connected by a thin rod of length L and negligible mass with the 4m mass midway between the m and 2m masses, as shown. If this composite object is rotated around an axis perpendicular to the rod and through a point midway between the 4 m and 2 m masses, what is its moment of inertia?
52mL2
1516mL2
716mL2
7mL2

Answer

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Answer

Final Answer: The moment of inertia of the composite object is 2116mL2. However, this option is not listed in the given choices. There might be a mistake in the problem or in my calculations.

Steps

Step 1 :The moment of inertia of a point mass is given by I=mr2, where m is the mass and r is the distance from the axis of rotation. In this case, we have three point masses at different distances from the axis of rotation. We can calculate the moment of inertia of each mass and then add them up to get the total moment of inertia.

Step 2 :The 4m mass is at a distance of L4 from the axis of rotation, the 2m mass is at a distance of L2 from the axis of rotation, and the m mass is at a distance of 3L4 from the axis of rotation.

Step 3 :Calculate the moment of inertia for each mass: I4m=4m(L4)2=mL24, I2m=2m(L2)2=mL22, and Im=m(3L4)2=9mL216.

Step 4 :Add up the moments of inertia to get the total moment of inertia: Itotal=I4m+I2m+Im=mL24+mL22+9mL216=21mL216.

Step 5 :Final Answer: The moment of inertia of the composite object is 2116mL2. However, this option is not listed in the given choices. There might be a mistake in the problem or in my calculations.

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