A rope of negligible mass is wrapped around a 225-kg solid cylinder of radius 0.400 m. the cylinder is suspended several meters off the ground with its axis oriented horizontally, and turns on that axis without friction. (a) if a 75.0-kg man takes hold of the free end of the rope and falls under the force of gravity, what is his acceleration? (b) what is the angular acceleration of the cylinder? (c) if the mass of the rope were not neglected, what would happen to the angular accelera- tion of the cylinder as the man falls?

Here [tex]r[/tex] is the radius of the cylinder, [tex]F[/tex] is the force applied by man ( produce of mass of man and gravitational force) and [tex]I[/tex] is the inertia (half of the product of mass of cylinder and square of radius of it).

Put the values,

[tex]a_c=\dfrac{r\times m_m g}{\dfrac{m_c r^2}{2} }\\a_c=\dfrac{2m_m g}{m_c r}\\ a_c=\dfrac{2\times75\times 9.81}{225\times 0.4}\\ a_c=16.35[/tex]

The acceleration of the cylinder is 16.35 rad per second squared.

As the acceleration of the man is product of the angular acceleration of the cylinder and its radius. Thus,

[tex]a_m=a_c\times r\\a_m=16.35\times0.04\\a_m=6.54[/tex]

Hence the acceleration of the man is 6.54 meter per second squared.

(b) The angular acceleration of the cylinder-

The angular acceleration of the cylinder can be given as,

[tex]a_c=\dfrac{rF}{I}[/tex]

Here [tex]r[/tex] is the radius of the cylinder, [tex]F[/tex] is the force applied by man ( produce of mass of man and gravitational force) and [tex]I[/tex] is the inertia (half of the product of mass of cylinder and square of radius of it).

Put the values,

[tex]a_c=\dfrac{r\times m_m g}{\dfrac{m_c r^2}{2} }\\a_c=\dfrac{2m_m g}{m_c r}\\ a_c=\dfrac{2\times75\times 9.81}{225\times 0.4}\\ a_c=16.35[/tex]

Hence the angular acceleration of the cylinder is 16.35 rad per second squared.

(c) if the mass of the rope were not neglected, then the angular acceleration of the cylinder as the man falls-

The mass of the rope will increase the inertia of the cylinder. This will result in more angular acceleration of the cylinder.

Hence if the mass of the rope were not neglected, then the angular acceleration of the cylinder as the man falls will be more.

Hence,

The acceleration of the man is 6.54 meter per second squared.
The angular acceleration of the cylinder is 16.35 rad per second squared.
if the mass of the rope were not neglected, then the angular acceleration of the cylinder as the man falls will be more.

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