What Is the Acceleration of Block B in a Two-Mass Pulley System?

[Neolight]

Homework Statement

In the figure shown if the system is released from rest, acceleration of block B will be
(a). g/2
(b). g/3
(c). 2g/3
(d). g/7

Homework Equations

ma= F[net]
taking the direction of motion as positive

The Attempt at a Solution

So my attempt was to first find tension in the string connected to Mass 3M from
3Ma= 3Mg-T
T= 3M(g-a)

now the tension in the second string connected to the M say T1 is half of T ( am i correct?)
so
T1= T/2= 3Mg/2 - 3Ma/2

now equation of motion for the mass M

Ma= T1-Mg
Ma= M(3g/2 -3a/2 -g)

a = 3g/2 - g - 3a/2
a + 3a/2 = 3g/2 -g
5a/2= g/2
∴ a= g/5

where did i go wrong , because i looked up the answer and it is g/7
please help...
[/B]

 

[Vibhor]

Homework Statement

In the figure shown if the system is released from rest, acceleration of block B will be
(a). g/2
(b). g/3
(c). 2g/3
(d). g/7

Homework Equations

ma= F[net]
taking the direction of motion as positive
View attachment 207210

The Attempt at a Solution

So my attempt was to first find tension in the string connected to Mass 3M from
3Ma= 3Mg-T
T= 3M(g-a)

now the tension in the second string connected to the M say T1 is half of T ( am i correct?)
so
T1= T/2= 3Mg/2 - 3Ma/2

now equation of motion for the mass M

Ma= T1-Mg
Ma= M(3g/2 -3a/2 -g)

a = 3g/2 - g - 3a/2
a + 3a/2 = 3g/2 -g
5a/2= g/2
∴ a= g/5

where did i go wrong , because i looked up the answer and it is g/7
please help... [/B]

Acceleration of block B is not 'a' i.e it is not equal to that of block A .

But they have a simple relationship .

 

[Neolight]

Acceleration of block B is not 'a' i.e it is not equal to that of block A .

But they have a simple relationship .

can you please tell me the relationship between the two accelerations?

 

[Vibhor]

can you please tell me the relationship between the two accelerations?

If the lower pulley goes down by a distance 'x' how much does block B go down ?

 

[Neolight]

If the lower pulley goes down by a distance 'x' how much does block B go down ?

since the string isn't directly connected to the pulley so they won't travel the same distance, the string goes along the circumference of the pulley, this is as far as i can think sorry I'm really bad at this type of thinking

 

[Vibhor]

since the string isn't directly connected to the pulley so they won't travel the same distance, the string goes along the circumference of the pulley, this is as far as i can think sorry I'm really bad at this type of thinking

No problem .

Suppose you hold block B i.e you do not allow it to move .Now let lower pulley go down by distance 'x' , how much string length of string whose one end is fixed at ground and other end connects block B , goes slack/loose ?

Hint: Carefully look at both the left and right parts of string going over lower pulley .

 

[Neolight]

No problem .

Suppose you hold block B i.e you do not allow it to move .Now let lower pulley go down by distance 'x' , how much string length of string whose one end is fixed at ground and other end connects block B , goes slack/loose ?

Hint: Carefully look at both the left and right parts of string going over lower pulley .

its half the circumference (πr) so if the pulley moves up say x distance the mass B will move πr +x ?

 

[Vibhor]

its half the circumference (πr) so if the pulley moves up say x distance the mass B will move πr +x ?

No.

You do not need to worry about the part that lies on the pulley circumference . Why ?
Because before pulley moves down , πr length is over the pulley and after pulley moves down , same length πr is on the pulley circumference , so net change in the length of string as far as part that lies on the string is zero .

So forget about the part of string that lies on pulley circumference .

Just look at left and right parts of string .Pick a pen and paper and make a rough sketch . Without moving block B , move lower pulley down by a distance 'x' .

How much string length of left part gets loosened ?

How much string length of right part gets loosened ?

What is the total length of string that gets slack/loosened ?

 

[Neolight]

No.

You do not need to worry about the part that lies on the pulley circumference . Why ?
Because before pulley moves down , πr length is over the pulley and after pulley moves down , same length πr is on the pulley circumference , so net change in the length of string as far as part that lies on the string is zero .

So forget about the part of string that lies on pulley circumference .

Just look at left and right parts of string .Pick a pen and paper and make a rough sketch . Without moving block B , move lower pulley down by a distance 'x' .

How much string length of left part gets loosened ?

How much string length of right part gets loosened ?

What is the total length of string that gets slack/loosened ?

hahah i get it now 2x

so acceleration of block B will be two times the accelaration of the pulley

i have done the calculation using this result and now the answer is g/7 ... thanks for your help , i really appreciate it

 

[Vibhor]

Well done !

 

[TSny]

i have done the calculation using this result and now the answer is g/7

Is this what you get for the acceleration of block B, or is it the acceleration of block A?

 

[Neolight]

Is this what you get for the acceleration of block B, or is it the acceleration of block A?

It's for block B

 

[TSny]

It's for block B

Maybe I'm making a mistake, but I get that block A has the acceleration of g/7.
Do you agree with the following equations?

3Mg - T_A = 3Ma_A

T_B - Mg = Ma_B

T_A = 2T_B

a_A = a_B / 2

 

[Neolight]

Maybe I'm making a mistake, but I get that block A has the acceleration of g/7.
Do you agree with the following equations?

3Mg - T_A = 3Ma_A

T_B - Mg = Ma_B

T_A = 2T_B

a_A = a_B/2

Are you taking the acceleration of the block A and B as the same?

 

[TSny]

Are you taking the acceleration of the block A and B as the same?

No. See the last equation I wrote.

 

[Neolight]

No. See the last equation I wrote.

The acceleration of block B will be two times that of block A while tension of Block B will be half that of block A

 

[TSny]

The acceleration of block B will be two times that of block A while tension of Block B will be half that of block A

Yes, that agrees with the 3rd and 4th equations that I wrote.

 

[Neolight]

Yes, that agrees with the 3rd and 4th equations that I wrote.

I have done the math again ... And the result is

Acceleration for block A is g/7 and for B is 2g/7

Correct?

I made a mistake in a my previous math

 

[TSny]

Acceleration for block A is g/7 and for B is 2g/7

OK, good. That's what I get, too.