# Kinematics and dynamics

The weights of the blocks A, B and C, shown in figure worth, 89N, 44,5 N and 44,5 N, respectively. The blocks are initially at rest. Knowing that B travels 2,44 m in 2s, determine (a) the module of F and (b) the tension on the rope AD. Discard the masses of the pulleys and friction.

(a) 7,37 N
(b) 92,7 N

My solution

Block B
ZFy = mB . aB
T - F - mB.g = mB . aB

Block C
ZFy = mC . aC
-mC . g + T = mC . aC

aB = -aC

I can not find the answer of the book

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rl.bhat
Homework Helper
Using the formula s = 1/2*at^2, find the acceleration of mb.

tiny-tim
Homework Helper
Hi Apprentice123!
aB = -aC

Nooo … not this time!

because … ?

Hi Apprentice123!

Nooo … not this time!

because … ?

Because ? I do not know
I have:

x = 1/2 * aB * t^2
aB = 1.22 m/s^2

Block B
T - F - PB = mB * aB
F = T - 50,034

Block C
-PC + T = mC * aC
T = 44,5 + 4,5361aC

How to calculate aC?

rl.bhat
Homework Helper
For block B, it should be
F + mB*g - T = mB*aB

For block B, it should be
F + mB*g - T = mB*aB

Yes, thanks. How do I calculate aC?

tiny-tim
Homework Helper
Hi Apprentice123!
Yes, thanks. How do I calculate aC?

General tip: if you can't see clearly what to do, then give everything names (or letters)

call the lengths of the ropes above A B C and D a b c and d, respectivley.

Then the height of B is -(b + d), also b + c is constant, and so is a + d.

Carry on from there …

andrevdh
Homework Helper
What is this module, first time for me? Don't you think it is the resultant force acting on B?

rl.bhat
Homework Helper
What is this module, first time for me? Don't you think it is the resultant force acting on B?
No. If you observe the weights of A, B and C , You can see that they are in equilibrium. To start the motion, a force F is applied on B.

andrevdh
Homework Helper
So the accelerations of B and C will differ due to the fact that the acceleration of pulley D "adds to" the motion of B and "subtracts from" the motion C?

I have a very similar question that I'm having problems solving,
For mass B,
$$\downarrow$$F + 44.5 - T = 4.45a
and a=1.22ms-2

For C,
$$\uparrow$$ T-44.5 = ma'

At first I thought a=a' and I tried working with it but I get a different answer,and after reading the other posts,I'm not sure they(the accelerations) are the same,but I don't understand why they are not either and how I'm supposed to calculate it.

Hope my problem's clear and I hope someone can help me

tiny-tim
Homework Helper
I have a very similar question …

At first I thought a=a' and I tried working with it but I get a different answer,and after reading the other posts,I'm not sure they(the accelerations) are the same,but I don't understand why they are not either and how I'm supposed to calculate it.

Hi leena19!

See post #7 …

The height of B is -(b + d), and so the accleration of B is -(b'' + d''), while for C it's … ?

Hello tiny-tim

See post #7 …

I've seen it,sadly I don't understand it :(

I understand ,why b+c and c+d is constant.Also,why a+b+d and a+c+d is constant,cause the length of the string's constant.
but i don't get why the height of B is negative(b+d),are we taking displacements here?
The height of B is -(b + d), and so the accleration of B is -(b'' + d''), while for C it's … ?
i don't know.is it -(c+d) or (c+d) ? and you get aB = -(b'' + d'') ,by double differentiaion,is it?
When i use,
a+b+d , i get aA + aB = 0,and aA = -aB
and for a+c+d,i get aA = -aC ,which can't be right,right?
I've never dealt with 2 pulley problems before,but I've done alot of single pulley problems,but we've never had to consider the heights there to find the acceleration,so I'm a bit lost here.

THANK YOU

This is an old thread I found on double pulleys

https://www.physicsforums.com/archive/index.php/t-258463.html"

I understand here why the acceleration is doubled for m1,as explained in the thread,but
I don't get why we can't apply the same principle here.
I also can't understand how the accelerations of B and C are different,in this system of 3 pulleys?and what the heights of the weights' got to do with it?
I think the acceleration of A is 1/2 that of B &/or 1/2 the acceleration of C?

I hope someone can explain to me the motion of this pulley system.

Thank you

Last edited by a moderator:
andrevdh
Homework Helper
Accelerations are vectors, and since the motion here is one dimensional one can add/subtract the various accelerations that the components in this system is experiencing. This also follows from Newton's second law, that is each force in the resultant equation for each mass has its own contribution to the resultant acceleration.

That being said one can see that D and A will share the same acceleration. D's acceleration is downwards in this situation. B and C would have had the same acceleration if it were not for the acceleration of D.

B accelerates downwards and C upwards. This mean that their common acceleration will be altered as follows:

The acceleration of D will subtract from their common acceleration for C and add for the acceleration of B.

Hi all,

I tried solving this. And got a different answer. Below is what I did.

Let us assume that acceleration of a block is positive in downward direction (as B moves down). Let the tension in the rope A-D be T1 and that in rope BC be T2.

Note that the system is an accelerating system. This implies that pulley D is going down with acceleration aA. Also with respect to D, accelerations of B and C need to be equal but in opposite directions. If accelerations of B and C wrt to D are aBD and aCD respectively, then

aB=aA+aBD aB is acceleration of B with respect to fixed point of pulleys between A and D
aC=aA+aCD aC is acceleration of C with respect to fixed point of pulleys between A and D

aCD=-aBD
Hence, aC=aA-aBD

For block B,
F+mB*g-T2=mB*aBD

For block C,
mC*g-T2=mC*aCD

For pulley D (as its massless),
2*T2=T1

For block A,
T1-mA*g = mA*aA

Now, we have been given the values of the weights in N (i.e. in the form of mX*g). Also given, is the displacement of B and the duration for the displacement.

aB=2*s*t^2 gives us aB=1.22 m/s^2

assuming g=10 m/s^2 gives us mA=8.9 kg, mB=mC=4.45 kg, we get for block B,

From here on, mA=2m and mB=mC=m. Adding equations obtained for block B and block C,
F + 2*m*g - 2*T2 = 0 as aCD = - aBD
F + 2*m*g = T1 ------------- (1)

Subtracting them gives,
F = 2*m*aBD ------------- (2)

From equation for block A we have,
T1 - 2*m*g = 2*m*aA
T1 = 2*m*aA + 2*m*g -------------- (3)

Using (3) in (1),
F = 2*m*aA

Compare this (2),
2*m*aA = 2*m*aBD
aBD = aA (:D moment)

As aB = aA + aBD,
aA = 0.5 * aB = 0.61 m/s^2

F = 2*m*aA = 54.29 N
T1 = F1 + 2*m*g = 143.29 N

If someone can find my mistake kindly assist. Already grateful to those who can help.

F = 2*m*aA = 54.29 N
T1 = F1 + 2*m*g = 143.29 N

I made a mistake here. The values I got were,

F = 5.429 N
T1 = 5.429 + 89 = 94.429 N

Still very different from the answer. :(

When you apply a force F on B, it moves down a distance b. It pulls A and C up with distance a and c respectively.
So b should be equal to a + c

Thank you rl.bhat.This is all that I needed to know.
But just to makesure I've understood it fully, if we were to take the distance interms of d when B is moving downwards,would it be like this?
b-c+d = 0
b = c-d
since -d = +a,
b = c+a, or
aB = aC + aA

andrevdh
Homework Helper
anant,

..........

For block B,
F+mB*g-T2=mB*aBD

For block C,
mC*g-T2=mC*aCD

......

you should use the accelerations aB and aC here, that is the forces are such so that the masses obtained these resultant accelerations - Newton's second law.

you should use the accelerations aB and aC here, that is the forces are such so that the masses obtained these resultant accelerations - Newton's second law.

Still not with you. Could you please explain?

rl.bhat
Homework Helper
Thank you rl.bhat.This is all that I needed to know.
But just to makesure I've understood it fully, if we were to take the distance interms of d when B is moving downwards,would it be like this?
b-c+d = 0
b = c-d
since -d = +a,
b = c+a, or
aB = aC + aA
I have observed a flaw in my post #18. Here is a new approach.
The given system contains two ropes. Each rope has two segments. when the force F is applied, all the segments will change their lengths. Note down the changes in the segments of a rope,and equate the net change to zero. Here we go.
When a force F is applied on B, the displacements of different masses and pulley D are:
B down by b, C up by c, D down by d and A up by a.
Now let us consider change in length of rope connecting Band C.
Since D moves down by d, both B and C will move down by d. Hence
net change in segment B = b-d
net change in segment C = -c-d
So the net change in the rope = b-d-c-d = 0. Or 2d = b - c.
And in rope connecting A and D, a = d.
Hence 2aA = aB - aC. Or aB = 2aA + aC
(F + mB*g - T)/mB = 2( 2T - mA*g)/mA + ( T - mC*g)
1.22 = 2(2T - 89)/8.9 + ( T - 44.5 )/4.45
= (6T - 267)/8.9
1.22*8.9 = 6T - 267
6T = 277.9
Tension in AD = 2T = 92.62 N.

The given system contains two ropes. Each rope has two segments. when the force F is applied, all the segments will change their lengths. Note down the changes in the segments of a rope,and equate the net change to zero. Here we go.
When a force F is applied on B, the displacements of different masses and pulley D are:
B down by b, C up by c, D down by d and A up by a.
Now let us consider change in length of rope connecting Band C.
Since D moves down by d, both B and C will move down by d.

I understand all of the above,rl.bhat.
It is this part that I find difficult to understand,
net change in segment B = b-d
net change in segment C = -c-d
We're taking displacements here,right?I think that's what you mean by net change in segment?so
When I, tried taking displacements,I end up with,
$$\uparrow$$ B= -b +d
C = c-d
net change in rope = -b - d + c - d = 0
and 2d = c - b ,which I know is wrong ,cause when I tried do it like this,I don't get the answer,but I really don't know why it's wrong? :(

rl.bhat
Homework Helper
I understand all of the above,rl.bhat.
It is this part that I find difficult to understand,

We're taking displacements here,right?I think that's what you mean by net change in segment?so
When I, tried taking displacements,I end up with,
$$\uparrow$$ B= -b +d
C = c-d
net change in rope = -b - d + c - d = 0
and 2d = c - b ,which I know is wrong ,cause when I tried do it like this,I don't get the answer,but I really don't know why it's wrong? :(
You are applying the force to B. So the length of the segment B is increasing. There fore b should be positive. Similarly C is going up. So the length of the segment C is decreasing. Therefore c should be negative. Change of the length of the segment is measured as the final length - initial length.

andrevdh
Homework Helper
anant,

Your accelerations aBD and aCD is the accelerations of B and C w.r.t. D, and since D is also accelerating these are not the resultant accelerations. Newton's second law states that the sum of the forces acting on a body is equal to the resultant acceleration times its mass, which is your accelerations aB and aC.

Also the accelerations aBD and aCD will be such that

aBD = - aCD

" aB=aA+aBD aB is acceleration of B with respect to fixed point of pulleys between A and D
aC=aA+aCD aC is acceleration of C with respect to fixed point of pulleys between A and D"

should rather be

aB=aA+aBD
aC=aA-aBD

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You are applying the force to B. So the length of the segment B is increasing. There fore b should be positive. Similarly C is going up. So the length of the segment C is decreasing. Therefore c should be negative. Change of the length of the segment is measured as the final length - initial length.

I'm sorry,rl.bhat.I still don't get it.
for B
final length= b+d
initial lenght = b
so substracting gives me -b

and for C,
final length = d-c
initial length = c
and subtacting to find the net change in segment gives me = d-2c ?