Please help I am in a hurry!! Consider the following four objects: a hoop, a solidsphere, a hollow sphere, a flat disk. Each of the objects hasa mass

asked 2021-01-08
Please help I am in a hurry!!
Consider the following four objects: a hoop, a solidsphere, a hollow sphere, a flat disk. Each of the objects hasa mass M and a radius R. The axis of rotation passes throughthe center of each object, and is perpedicular to the plane of thehoop and the plane of the flat disk. Which object requiresthe largest torque to give it the same angular acceleration. I need to justify my answers and rank the objects starting with theone that requires the largest torque.

Answers (1)


ok, Net Torque\( = Inertia \times\)\(\displaystyle\alpha\)
Inertia Hoop= \(\displaystyle{M}{R}^{{{2}}}\)
Inertia Solid Sphere= \(\displaystyle{\left({\frac{{{2}}}{{{5}}}}\right)}\cdot{M}{R}^{{{2}}}\)
Inertia Hollow Sphere= \(\displaystyle{\left({\frac{{{2}}}{{{3}}}}\right)}\cdot{M}{R}^{{{2}}}\)
Inertia Disk= \(\displaystyle{\left({\frac{{{1}}}{{{2}}}}\right)}\cdot{M}{R}^{{{2}}}\)
If all 4 objects have equal alphas(angular acceleration), then theone with the highest Inertia will have the greatest torque, sincethey all involve \(MR^{2}\), just compare the coefficients...
\(\displaystyle{1}{>}{\left({\frac{{{2}}}{{{3}}}}\right)}{>}{\left({\frac{{{1}}}{{{2}}}}\right)}{>}{\left({\frac{{{2}}}{{{5}}}}\right)}\).....thus the rank would be

\(Hoop>Hollow\ Sphere>Disk>Solid\ Sphere\)
The one that requires the highest amount of torque is: a hoop, thin/hollow sphere, disc, solid sphere.
The reason for this is that the moment of inertia is the highestfor the hoop, and then goes down as follows moment of inertia is know as I, and the equation is \(I=constant\times M\times R^{2}\), the constant varies for the type of object and itcan be found in most physics books.

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