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songoku

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- Homework Statement
- Sphere A of mass 2 kg is connected to object B (4 kg) through a pulley (0.5 kg). If sphere A rolls without slipping, find the acceleration of the system.

(Radius of pulley = 20 cm)

- Relevant Equations
- Στ = I . α

ΣF = m.a

There is no friction mentioned by the question so I assume the plane is frictionless but can the sphere roll without slipping if there is no friction?

This is my attempt:

Equation of translation motion of object A (assuming A moves upwards):

T

_{A}- W

_{A}sin θ = m

_{A}. a

_{COM (A)}

T

_{A}= m

_{A}. a

_{COM (A)}+ W

_{A}sin θ ...(1)

Equation of rotational motion of object A:

T

_{A}. R = ##\frac{2}{5}## m

_{A}R

^{2}. ##\frac{a_{tangential(A)}}{R}##

T

_{A}= ##\frac{2}{5}## m

_{A}. a

_{tangential (A)}...(2)

Since A rolls without slipping → a

_{COM (A)}= a

_{tangential (A)}

Equating (1) and (2), I get a

_{COM (A)}= - 13 ms

^{-2}→ is this value possible? bigger than acceleration of free fall?

So this means sphere A rolls downwards, but the acceleration of the point of contact between the sphere and inclined plane (let say point X) is not zero so the sphere is not rolling without slipping with respect to point X. Does this mean the question refers the motion of rolling without slipping actually with respect to the point where the tension acts on the sphere / top part of the sphere (let say point Y)?

But the acceleration of point Y on sphere A should be the same as the acceleration of the center of mass of object B so the acceleration of B is zero?

I am really confused. Where is my mistake?

And by "acceleration of system", what does the question refer to? Is it acceleration of center of mass of A or of B?

Thanks

EDIT: or maybe actually this question is about static condition? None is moving?

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