Velocity of a 1-kg block after it has dropped 0.54 m

[volcore]

Homework Statement

Two blocks are hung by a string draped over a pulley, a 1.0-kg block on the left and a 3.5-kg block on the right. The two blocks start out at rest and at the same height.

What is the change in the gravitational potential energy of the system of blocks and Earth when the 3.5-kg block has dropped 0.54 m ?

What is the velocity of the 1.0-kg block at this instant?
Express your answer with the appropriate units.

Relevant Equations

k=mgh
k=0.5mv^2

I calculated the system's change in gravitational potential energy through the equation k=mgh. I used this equation twice, once for the 3.5 kg block with k =(3.5)(9.8)(-0.54) and once for the 1.0kg block, k = (9.8)(1.0). I got -18.522J & 5.292. Is the change just -18.522 - 5.292 J?

Furthermore, to calculate the velocity, I tried using k = 0.5mv^2 with k = 5.292 & m = 1.0, getting a velocity of 3.3m/s, which is apparently wrong. What am I missing?

 

[PeroK]

I calculated the system's change in gravitational potential energy through the equation k=mgh. I used this equation twice, once for the 3.5 kg block with k =(3.5)(9.8)(-0.54) and once for the 1.0kg block, k = (9.8)(1.0). I got -18.522J & 5.292. Is the change just -18.522 - 5.292 J?

Furthermore, to calculate the velocity, I tried using k = 0.5mv^2 with k = 5.292 & m = 1.0, getting a velocity of 3.3m/s, which is apparently wrong. What am I missing?

The blocks form a coupled system. Are you you sure you can calculate their KE independently?

In what direction is the 1kg block moving? Up or down? Is it losing or gaining GPE? Do objects normally accelerate upwards under gravity?

Hint: imagine what would happen if the blocks were almost the same mass. How quickly would the system accelerate? How would that affect the velocity after they had moved a certain distance?

 

[volcore]

The blocks form a coupled system. Are you you sure you can calculate their KE independently?

I'm not too sure, my professor never mentioned anything about coupled systems yet, let alone if calculating their kinetic energy would be different or not.

In what direction is the 1kg block moving? Up or down? Is it losing or gaining GPE? Do objects normally accelerate upwards under gravity?

I'd assume the 1kg block is moving up. While it wouldn't be accelerating under gravity, won't it have the same acceleration as the 3.5kg block?

Hint: imagine what would happen if the blocks were almost the same mass. How quickly would the system accelerate? How would that affect the velocity after they had moved a certain distance?

Wouldn't the system accelerate slower, thus the velocity would be less in general?

 

[PeroK]

I'd assume the 1kg block is moving up. While it wouldn't be accelerating under gravity, won't it have the same acceleration as the 3.5kg block?Wouldn't the system accelerate slower, thus the velocity would be less in general?

Yes and yes. But you didn't take either of those factors into account.

The fact that they have the same acceleration is what it means to be coupled!

 

[volcore]

Yes and yes. But you didn't take either of those factors into account.

Sorry, I'm confused, in the equations I used to find the individual kinetic energies, I used the same acceleration and height, except I inverted it for the 3.5kg block to -0.54. since it's going down. Looking up, it looks like I missed writing that fact for the second equation, apologies. Should I have inverted the acceleration of the 3.5kg block since it's accelerating downwards, making it -9.8?

The fact that they have the same acceleration is what it means to be coupled!

Would that have any affect on finding the kinetic energy?

 

[PeroK]

Sorry, I'm confused, in the equations I used to find the individual kinetic energies, I used the same acceleration and height, except I inverted it for the 3.5kg block to -0.54. since it's going down. Looking up, it looks like I missed writing that fact for the second equation, apologies. Should I have inverted the acceleration of the 3.5kg block since it's accelerating downwards, making it -9.8?Would that have any affect on finding the kinetic energy?

Yes, because the lighter block is actually retarding the system.

 

[PeroK]

If you stick with your energy approach - which is a good idea - the lighter block is gaining GPE!

 

[volcore]

So should I still calculate the 2 individual kinetic energies separately?

 

[PeroK]

So should I still calculate the 2 individual kinetic energies separately?

The blocks move together. The light block is moving up. Which is impossible unless you consider it linked to the heavier block.

Things don't accelerate upwards under gravity.

It's a pity you can't set up an experiment to see what happens when two masses are linked around a pulley. If you don't get the idea from thinking about it, then you perhaps need to see a physical system like this in action.

Your solution is only valid if someone cuts the rope joining them. Then they both fall independently. But that's not what's happening here.

 

[volcore]

The blocks move together. The light block is moving up. Which is impossible unless you consider it linked to the heavier block.

Things don't accelerate upwards under gravity.

It's a pity you can't set up an experiment to see what happens when two masses are linked around a pulley. If you don't get the idea from thinking about it, then you perhaps need to see a physical system like this in action.

Your solution is only valid if someone cuts the rope joining them. Then they both fall independently. But that's not what's happening here.

So I'd need to calculate their kinetic energy together? would I just use the equation k=mgh with m = their combined mass of 4.5, g = 9.8 and the height = 0.54?

 

[PeroK]

So I'd need to calculate their kinetic energy together? would I just use the equation k=mgh with m = their combined mass of 4.5, g = 9.8 and the height = 0.54?

The lighter weight is going up, not down. It's gaining GPE. You need to calculate how much GPE the system loses.

 

[volcore]

The lighter weight is going up, not down. It's gaining GPE. You need to calculate how much GPE the system loses.

Isn't the equation for gravitational potential energy essentially still the same, U^g = mg h?

 

[PeroK]

Isn't the equation for gravitational potential energy essentially still the same, U^g = mg h?

So, what's your calculation for this problem?

 

[volcore]

Suppose you release a ball from a height of $1m$. When it reaches the height of $1.5m$, how fast is it travelling?

I'm not sure, since the ball is gaining height, wouldn't it decelerate?

 

[PeroK]

I'm not sure, since the ball is gaining height, wouldn't it decelerate?

It's starts from rest. It can't decelerate.

 

[volcore]

It's starts from rest. It can't decelerate.

Ah, I had my positives and negatives mixed up, it would accelerate due to gravity, right?

 

[PeroK]

Ah, I had my positives and negatives mixed up, it would accelerate due to gravity, right?

I'm reluctant to move from your energy-based approach to a force-based approach, because I think the energy-based approach is better. But, to answer this question:

No. Gravity is pulling the block downwards. It's being pulled upwards by tension in the string.

 

[volcore]

Wait, is your 0 at the top or bottom?

 

[PeroK]

Given the lack of progress, I better give you more than hints.

The large block is losing GPE and the smaller block is gaining GPE. The change in GPE of the system is:

$\Delta PE = -Mgh + mgh = -(M-m)gh = -(2.5kg)g(0.54m)$

From a force perspective, the larger block pulls the smaller block up via tension in the string; and the smaller block exerts a retarding force on the larger block by the same tension in the string.

Either way, you do not have the full gravitational acceleration in this system, because one mass is moving up.

 

[volcore]

Given the lack of progress, I better give you more than hints.

The large block is losing GPE and the smaller block is gaining GPE. The change in GPE of the system is:

$\Delta PE = -Mgh + mgh = -(M-m)gh = -(2.5kg)g(0.54m)$

From a force perspective, the larger block pulls the smaller block up via tension in the string; and the smaller block exerts a retarding force on the larger block by the same tension in the string.

Either way, you do not have the full gravitational acceleration in this system, because one mass is moving up.

That equation makes sense, but where do I go from there?

 

[PeroK]

What does that tell you about the KE of your system?

 

[PeroK]

That equation makes sense, but where do I go from there?

Here's the alternative force-based approach. Let $T$ be the tension in the string. The force on the large mass is downwards of magnitude:

$F_M = Mg - T$

And, the force on the smaller mass is upwards of magnitude:

$F_m = T - mg$

This shows you the acceleration more explicity. But, this way you have to figure out the value of $T$.

 

[volcore]

What does that tell you about the KE of your system?

That the system is gaining kinetic energy since its losing potential energy

 

[PeroK]

That the system is gaining kinetic energy since its losing potential energy

Yes. But, we knew that already. It's how much KE is gained that's important.

 

[volcore]

Yes. But, we knew that already. It's how much KE is gained that's important.

Wouldn't the system gain kinetic energy of around (2.5kg)g(0.54m) since that's how much potential energy its losing?

 

[kuruman]

I calculated the system's change in gravitational potential energy through the equation k=mgh. I used this equation twice, once for the 3.5 kg block with k =(3.5)(9.8)(-0.54) and once for the 1.0kg block, k = (9.8)(1.0). I got -18.522J & 5.292. Is the change just -18.522 - 5.292 J?

For the 3.5 kg block you multiplied (3.5)(9.8)(-0.54). That's good.
For the 1.0 kg block you multiplied (9.8)(1.0). You forgot to include the change in height (+0.54) m. When one block goes down by 0.54 m, the other goes up by the same amount because the connecting string does not stretch or shrink.

 

[volcore]

For the 3.5 kg block you multiplied (3.5)(9.8)(-0.54). That's good.
For the 1.0 kg block you multiplied (9.8)(1.0). You forgot to include the change in height (+0.54) m. When one block goes down by 0.54 m, the other goes up by the same amount because the connecting string does not stretch or shrink.

Yeah, I remembered to multiply by +0.54 in my actual calculation, but accidentally omitted it when posting the thread, my mistake.

 

[PeroK]

Wouldn't the system gain kinetic energy of around (2.5kg)g(0.54m) since that's how much potential energy its losing?

Yes it's got to be the same. So, how is that KE allocated between the blocks?

 

[volcore]

Yes it's got to be the same. So, how is that KE allocated between the blocks?

I'm not sure. Since I can't calculate their individual kinetic energies, I'm kind of lost. Would the heavier block gain more since it's presumably moving faster?

 

[vela]

I'm not sure. Since I can't calculate their individual kinetic energies, I'm kind of lost. Would the heavier block gain more since it's presumably moving faster?

Faster than what?

 

[volcore]

Faster than what?

Bad wording on my part, since kinetic energy is defined as 1/2 mv^2, wouldn't the 3.5kg block receive more of the system's kinetic energy since it has more mass, and both are moving at the same velocity?

 

[PeroK]

Bad wording on my part, since kinetic energy is defined as 1/2 mv^2, wouldn't the 3.5kg block receive more of the system's kinetic energy since it has more mass, and both are moving at the same velocity?

Yes, exactly.

 

[volcore]

Yes, exactly.

So where do I go from there?

Would this be right:
(2.5kg)g(0.54m) = 1/2 m1v1^2 + 1/2m2v2^2
1.35g = 3.75v1^2 + 0.5v2^2?

 

[PeroK]

So where do I go from there?

Would this be right:
(2.5kg)g(0.54m) = 1/2 m1v1^2 + 1/2m2v2^2
1.35g = 3.75v1^2 + 0.5v2^2?

What do you know about the relationship between $v_1$ and $v_2$?

Note that you should keep the units in these equations.