# 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
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## 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
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?

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

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
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 ?

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

Well done !

Neolight
TSny
Homework Helper
Gold Member
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?

Vibhor
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
Homework Helper
Gold Member
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 - TA = 3MaA

TB - Mg = MaB

TA = 2TB

aA = aB / 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 - TA = 3MaA

TB - Mg = MaB

TA = 2TB

aA = aB/2
Are you taking the acceleration of the block A and B as the same?

TSny
Homework Helper
Gold Member
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
Homework Helper
Gold Member
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
Homework Helper
Gold Member
Acceleration for block A is g/7 and for B is 2g/7
OK, good. That's what I get, too.

Neolight