- #1

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- Homework Statement
- Some packages of mass ##0.8 kg## are deposited on a conveyor belt, whose coefficient of static friction is ##0.4##. Determine:

A) The force exerted by the conveyor belt on each package a moment before and after passing by the point ##A##.

B) The angular coordinate in point ##B##, which is the point where the packages start to slide

C) The angular coordinate when the packages lose contact with the conveyor belt.

- Relevant Equations
- Newton's equations

A) Before passing by ##A## the forces acting are weight and normal force. Then, after passing by that point you have

(Tangential direction) ##W_t-Fr=m.a_t##

(Normal direction) ##-N+W_n=m.\omega ^2 .r##

Also, as the packages don't slide, we can say that ##a_t=0##

Are these ideas okay?

B) When the packages are about to slide, the static friction is maximum. So you have

##\hat t) W_t-Fr=0##

##W_t=Fr##

##W.sin(\alpha)=\mu N##

From ##\hat n## we know that ##N=mg.cos(\alpha)-m.\omega ^2 .r##

So we can replace with that

##mg.sin(\alpha)=\mu (mg.cos(\alpha)-m.\omega ^2 .r)##

The thing is that I don't have ##r## and I don't have ##\alpha##.

C) If they lose contact, then ##N=0## and

##\hat n) -N+W_n=m.\omega ^2 r##

##mg.cos(\alpha)=m.\omega ^2 r##

##cos(\alpha)=\frac{\omega ^2 r}{g}##

But I have the same problem that I had in ##B##.

(Tangential direction) ##W_t-Fr=m.a_t##

(Normal direction) ##-N+W_n=m.\omega ^2 .r##

Also, as the packages don't slide, we can say that ##a_t=0##

Are these ideas okay?

B) When the packages are about to slide, the static friction is maximum. So you have

##\hat t) W_t-Fr=0##

##W_t=Fr##

##W.sin(\alpha)=\mu N##

From ##\hat n## we know that ##N=mg.cos(\alpha)-m.\omega ^2 .r##

So we can replace with that

##mg.sin(\alpha)=\mu (mg.cos(\alpha)-m.\omega ^2 .r)##

The thing is that I don't have ##r## and I don't have ##\alpha##.

C) If they lose contact, then ##N=0## and

##\hat n) -N+W_n=m.\omega ^2 r##

##mg.cos(\alpha)=m.\omega ^2 r##

##cos(\alpha)=\frac{\omega ^2 r}{g}##

But I have the same problem that I had in ##B##.