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Memo

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- Homework Statement
- (the photo is below) The angle between the inclined plane and the surface is α. The coefficient of friction between block 2 and the inclined plane is k. Find the ratio inequality of m1 over m2 that the system remains still.

Answer: for m2 to go down: m1/m2>sina+k∗cosa

for m2 to go up: m1/m2<sina−k.cosa

For the system to remain balanced: sina−k.cosa<m1/m2<sina+k.cosa

Could somebody explain to me why the answer should be the above and not sina+k.cosa<m1/m2<sina−k.cosa?

Thank you very much!!!

- Relevant Equations
- I combined the two inequalities but it doesn't match the answer hints so I'd love it if someone could explain to me how that makes sense.

(sorry for my poor English)

+For m1 to go down, the Net force acting on m1 must be smaller than 0 -> T-m1*g<0 -> T<m1*g(1)

_Evaluating object 2 perpendicularly: N-Py2=0 -> N=m2*g*cosa

_Evaluating object 2 parallel: -Px-f+T=0 -> T=sina*g*m2 + k*m2*g*cosa (2)

_Insert 2 into 1: m1g > sina*g*m2 + k*m2*g*cosa -> m1 > m2*(sina+kcosa) -> m1/m2 > sina + kcosa

+For m1 to go up, the Net force acting on m1 must be greater than 0 -> T-m1*g>0 -> T>m1*g (3)

_Evaluating object 2 parallel: -Px+f+T=0 -> T=sina*g*m2 - k*m2*g*cosa (4)

_Insert 4 into 3: m1/m2 < sina - k*cosa

+For the system to remain still, the tension force must be within the range when the objects are on the verge of moving.

(so I combine the two inequalities and get sina + k*cosa < m1/m2 < sina - k*cosa but the answer hints isn't the same)

+For m1 to go down, the Net force acting on m1 must be smaller than 0 -> T-m1*g<0 -> T<m1*g(1)

_Evaluating object 2 perpendicularly: N-Py2=0 -> N=m2*g*cosa

_Evaluating object 2 parallel: -Px-f+T=0 -> T=sina*g*m2 + k*m2*g*cosa (2)

_Insert 2 into 1: m1g > sina*g*m2 + k*m2*g*cosa -> m1 > m2*(sina+kcosa) -> m1/m2 > sina + kcosa

+For m1 to go up, the Net force acting on m1 must be greater than 0 -> T-m1*g>0 -> T>m1*g (3)

_Evaluating object 2 parallel: -Px+f+T=0 -> T=sina*g*m2 - k*m2*g*cosa (4)

_Insert 4 into 3: m1/m2 < sina - k*cosa

+For the system to remain still, the tension force must be within the range when the objects are on the verge of moving.

(so I combine the two inequalities and get sina + k*cosa < m1/m2 < sina - k*cosa but the answer hints isn't the same)

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