# Acceleration of Block on an Apparatus

[SOLVED] Acceleration of Block on an Apparatus

## Homework Statement

See the attachment.

## The Attempt at a Solution

I am confused about whether the tensions of the two (left and right) parts of the line are the same or not. Since the three blocks are accelerating so it's not a equilibrium situation. Are the tensions the same on either line?

If block m3 were not there, and instead, m1 and m2 were connected I can use the equations below to solve for acceleration. But I am a bit thrown off by the addition of m3.

(1) m1*g-T = m1*a
(2) T-m2*g = m2*a

Where T is the tension of the either line. But how do with m3, the block on the table?

PS: HallsofIvy, ShootingStar, I appreciate your help with the previous problem.

#### Attachments

• apparatus.jpg
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Doc Al
Mentor
No, the two tensions are not the same. If they were, things wouldn't make much sense: The net force on m3 would be zero and thus no acceleration. Call the tensions T1 and T2 and rewrite your two equations. And add a third equation for m3.

No, the two tensions are not the same. If they were, things wouldn't make much sense: The net force on m3 would be zero and thus no acceleration. Call the tensions T1 and T2 and rewrite your two equations. And add a third equation for m3.

Okay. So let the left and right lines have tensions T1 and T2 respectively.

(1) m1*g-T1 = m1*a
(2) T2-m2*g = m2*a
(3) m3-(T1+T2) = m3*a

Would the above be correct?

$$m_3$$ is not a force!

Doc Al
Mentor
Okay. So let the left and right lines have tensions T1 and T2 respectively.

(1) m1*g-T1 = m1*a
(2) T2-m2*g = m2*a
(3) m3-(T1+T2) = m3*a

Would the above be correct?
Equation 3 is not quite right. What's the net force on m3? (Be sure to use a consistent sign convention for the acceleration in all three equations.)

(1) m1*g-T1 = m1*a
(2) m2*g-T2 = m2*a
(3) m3*g-(T1+T2) = m3*a

About the sign convention for acceleration, I thought I needed to change the sign because as block m1 is accelerating say, upward, then m2 must be accelerating downward?

(1) m1*g-T1 = m1*a
(2) m2*g-T2 = m2*a
(3) m3*g-(T1+T2) = m3*a

About the sign convention for acceleration, I thought I needed to change the sign because as block m1 is accelerating say, upward, then m2 must be accelerating downward?

If $m_1$ is accelerating upward then $$T_1-m_1\,g=m_1\,a$$ and reverse signs for $m_2$.

$m_3$ is moving on the table. What that tells you about it's weight $m_3\,g[/tex]? Doc Al Mentor (1) m1*g-T1 = m1*a (2) m2*g-T2 = m2*a (3) m3*g-(T1+T2) = m3*a Equation 3 is still wrong. The weight--which acts vertically, not horizontally--is balanced by the normal force from the table. Only consider the horizontal forces on m3. Looks like you changed equations 1 and 2, so they need fixing now. (They were fine before!) About the sign convention for acceleration, I thought I needed to change the sign because as block m1 is accelerating say, upward, then m2 must be accelerating downward? Previously, in your first two equations you chose to have m1 accelerate up and m2 down when a is positive. Nothing wrong with that. Use that same convention when analyzing m3. [itex]m_3$ is moving on the table. What that tells you about it's weight $m_3\,g[/tex]? Other than that the net vertical force on m3 is zero, I'm not sure what else I can extrapolate. What Doc Al said. "consider the horizontal forces on m3" Equation 3 is still wrong. The weight--which acts vertically, not horizontally--is balanced by the normal force from the table. Only consider the horizontal forces on m3. Looks like you changed equations 1 and 2, so they need fixing now. (They were fine before!) Previously, in your first two equations you chose to have m1 accelerate up and m2 down when a is positive. Nothing wrong with that. Use that same convention when analyzing m3. Oh, m3*g is the vertical force. The horizontal force is just the tension. (1) m1*g-T1 = m1*a (2) T2-m2*g = m2*a (3) (T1+T2) = -m3*a The first two are ok! For the 3rd: are the tensions in the same direction? and why -a? Doc Al Mentor (3) (T1+T2) = -m3*a Do both tensions pull in the same direction? (1) m1*g-T1 = m1*a (2) T2-m2*g = m2*a (3) T1-T2 = m3*a The tensions are not in the same direction. I suppose because, I assumed +a applied to block m2 then +a must apply to m3 also. Although I have to admit I'm still not quite clear about this concept because the direction of acceleration of block m3 isn't in the direction of m2, the -y direction, rather it's in the +x direction. Every block moves at it's "own" axe. If you assume that [itex]m_1$ is going down then $m_2$ must go up and $m_3$ must move to the left. Thus the "+x direction" is to the left.

As small recipe is that when you are dealing with a system of mases like this one is that you can treat it as "one boby" system. All the internal forces must equal to zero (Newton's 3rd law) and you are left with the externals i.e. the two weight's.
Thus if you assume that $m_1$ is going down

$$m_1\,g-m_2\,g=(m_1+m_2+m_3)\,a$$

To put it in other words when you add all the equations the tensions must disapper!

Doc Al
Mentor
(1) m1*g-T1 = m1*a
(2) T2-m2*g = m2*a
(3) T1-T2 = m3*a
Good.

The tensions are not in the same direction. I suppose because, I assumed +a applied to block m2 then +a must apply to m3 also. Although I have to admit I'm still not quite clear about this concept because the direction of acceleration of block m3 isn't in the direction of m2, the -y direction, rather it's in the +x direction.
The way to understand this is to realize that since the blocks are all connected by strings they are constained to move with the same magnitude of acceleration. And if m1 moves down, then m2 must move up, and m3 must move to the left. This constraint is applied in your equations by using the same letter (a) to represent the magnitude of the acceleration, and by using a consistent sign convention.

Every block moves at it's "own" axe. If you assume that $m_1$ is going down then $m_2$ must go up and $m_3$ must move to the left. Thus the "+x direction" is to the left.

As small recipe is that when you are dealing with a system of mases like this one is that you can treat it as "one boby" system. All the internal forces must equal to zero (Newton's 3rd law) and you are left with the externals i.e. the two weight's.
Thus if you assume that $m_1$ is going down

$$m_1\,g-m_2\,g=(m_1+m_2+m_3)\,a$$

To put it in other words when you add all the equations the tensions must disapper!

I see that your equation was just the result of summing the equations 1,2, and 3. But how were you able to come up with it directly?

As I understand, the left side (m1+m2+m3)*a is the acceleration of the entire system of the 3 blocks. For the left side of the equation, what about m3*g? Did you not include it because m3 is accelerating along the table? I'd appreciate if you could clarify. Thanks!

Consider a system of N bloks. The net force on the whole system is

$$\sum\vec{F}_{external}+\sum\vec{F}_{internal}=m_{total}\,\vec{a}$$

The sum $$\sum\vec{F}_{internal}$$ equals zero by the Newton's 3rd law, thus all the tensions are gone away!

The sum $$\sum\vec{F}_{external}$$ has the three weights and the normal force on $m_3$ which cancels $m_3\,g$.

Thus you are left with the equation I posted!

Is this clear enough? My English are pretty poor!

Good.

The way to understand this is to realize that since the blocks are all connected by strings they are constained to move with the same magnitude of acceleration. And if m1 moves down, then m2 must move up, and m3 must move to the left. This constraint is applied in your equations by using the same letter (a) to represent the magnitude of the acceleration, and by using a consistent sign convention.

Doc Al, that explanation cleared it up for me I think.

So one direction is negative and the other is positive. I can ascribe a negative sign to the entire movement involving m1 moving down, m2 moving up, and m3 moving to the left, and vice versa. In other words, essentially there are only two directions: positive and negative. Is this reasoning correct?

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Doc Al
Mentor
So one direction is negative and the other is positive. I can ascribe a negative sign to the entire movement involving m1 moving down, m2 moving up, and m3 moving to the left, and vice versa. In other words, essentially there are only two directions: positive and negative. Is this reasoning correct?
Right: There are only two possibilities for the acceleration. I think you've got the idea.

Here's how I would put it: The key is that you used a consistent sign convention for the acceleration, one that reflects how the blocks are actually constrained to move. You chose a convention such that if the acceleration turns out (when you solve for it) to be positive, that means that m1 is accelerating down and m2 is accelerating up; but if the acceleration turns out to be negative, just the opposite.

Right: There are only two possibilities for the acceleration. I think you've got the idea.

Here's how I would put it: The key is that you used a consistent sign convention for the acceleration, one that reflects how the blocks are actually constrained to move. You chose a convention such that if the acceleration turns out (when you solve for it) to be positive, that means that m1 is accelerating down and m2 is accelerating up; but if the acceleration turns out to be negative, just the opposite.

Doc Al, I'm sorry to make you explain this again but you keep stressing that the acceleration sign conventions are consistent I cant help but think that they really aren't. This might be a silly question, if I just multiple through by (-1) for eq. 1 then I am writing the acceleration as negative aren't I?

(1) m1*g-T1 = m1*a // (1) T1-m1*g = -m1*a
(2) T2-m2*g = m2*a
(3) T1-T2 = m3*a

I recognize that the internal forces must ultimately all cancel but I guess I don't understand why the tensions T1 and T2 must be flipped around. Is it acting in a certain direction? Why would it not be correct to write eq. 3 as: T2-T1 = m3*a? They're still acting in difference directions right? If you could clarify this, I've think I have a complete grasp of the problem. Thank you so much.

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Doc Al
Mentor
Doc Al, I'm sorry to make you explain this again but you keep stressing that the acceleration sign conventions are consistent I cant help but think that they really aren't.
In order to use the same letter (a) to represent the acceleration of each mass, you need to be consistent in how you write your equations. I'll explain what that means down below.
This might be a silly question, if I just multiple through by (-1) for eq. 1 then I am writing the acceleration as negative aren't I?
No.
(1) m1*g-T1 = m1*a // (1) T1-m1*g = -m1*a
Just multiplying an equation by (-1) doesn't change the equation at all! Note: An equation is changed if its solution would be different. Those two equations have the exact same solution.
(2) T2-m2*g = m2*a
(3) T1-T2 = m3*a

I recognize that the internal forces must ultimately all cancel but I guess I don't understand why the tensions T1 and T2 must be flipped around. Is it acting in a certain direction? Why would it not be correct to write eq. 3 as: T2-T1 = m3*a? They're still acting in difference directions right? If you could clarify this, I've think I have a complete grasp of the problem. Thank you so much.
Realize that T1-T2 = m3*a and T2-T1 = m3*a are not the same, since they would have different solutions.

I'm going to show you the thinking I go through when I write the equations for this problem. Maybe that will help you.

I look at the problem and realize that since the blocks are connected they are constrained to move together with the same magnitude of acceleration. Until we plug in actual numbers for the masses, I have no idea which way they will accelerate. But it doesn't matter, as long as I am "consistent" with the way things must actually move. (By that I mean: If I assume m1 moves down, I must assume m2 moves up. Otherwise I'll get nonsense and will not be able to solve the equations.)

Since I don't know which way things will actually move, I'll just pick a direction at random. I will say that the acceleration of:
- m1 has magnitude "a" and goes down
- m2 has magnitude "a" and goes up
- m3 has magnitude "a" and goes to the left (towards m1)
Since these are all consistent I can use the same letter ("a") to represent the equations. If I happened to have picked the correct direction for the acceleration, when I solve for "a", "a" will be positive; If I happened to pick the wrong direction, "a" will turn out to be negative. No problem!

Now I will write the equations for each mass. I always use the convention that up is + and down is negative, but it doesn't matter:
for m1: T1 - m1g = m1(-a) = -m1a; which is equivalent to: m1g - T1 = m1a.
(Note that since the acceleration is "a" downward, I call it -a.)
for m2: T2 - m2g = m2a (since the acceleration of m2 is "a" up I call it +a.)

For the third equation, since it's moving horizontally, I'll chose the convention that "right" is + and left is negative. But it doesn't matter!
for m3: -T1 + T2 = m3(-a); which is equivalent to: T1 - T2 = m3a.
(Note that since T1 acts to the left, I call it -T1; and since the acceleration is also to the left, I call it -a.)

(I always treat T1, T2, and a as if they are positive numbers.)

I hope that helps a bit. I assure you, it's easier than it looks.

For the third equation, since it's moving horizontally, I'll chose the convention that "right" is + and left is negative. But it doesn't matter!
for m3: -T1 + T2 = m3(-a); which is equivalent to: T1 - T2 = m3a.
(Note that since T1 acts to the left, I call it -T1; and since the acceleration is also to the left, I call it -a.)

Doc Al, you're wonderful. This was precisely the part I had trouble grasping.

I read your explanation and many times. Now I understand. I have to say that I found it sort of interesting that eq. 3 comes out to be the same if I assumed the opposite you did, that is, "right" is is negative and "left" is positive. Keeping the initial assumption that m3 is moving to the left, it follows that acceleration is then positive, and the signs of T1 and T2 are hence also reversed. I come up with eq. 3 directly!
T1-T2=m3a

You have no idea how much that cleared things up for me. Thanks again.

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