You're welcome, Rémi! Good luck with your exercise.

In summary, the conversation discusses a problem involving two blocks connected by a string and released from rest with no friction. The goal is to find how far block M1 slides in time t. The approach involves using Newton's laws and the relation between x(t), v(t), and a(t) to determine the tension and acceleration of the blocks. The summary also notes a mistake made in assuming the magnitude of T_M2 and W_M2 were equal.
  • #1
Remi David
2
1
Hello !

Homework Statement


Two blocks M1 and M2 are connected by a string of negligible mass. If the system is released from rest, find how far block M1 slide in time ##t##. Neglect friction.

Diagram:

See Attached Image

Clue given in the manual:

If M1 = M2, then solution is ##x(t)= \frac{gt^2}{4}##

Homework Equations


[/B]
Newton's Laws:

##\sum_{}^{} \vec{F} = m\vec{a}##
## \vec{F}_{m1/m2} = -\vec{F}_{m2/m1} ##

Relation between ##x(t)##, ##\vec{v}(t)##, ##\vec{a}(t)##:

##\vec{v}(t) = \frac{dx}{dt}##
##\vec{a}(t) = \frac{dv}{dt}##

The Attempt at a Solution



So I started the exercise by separating the two masses M1 and M2, and I did a diagram with all the forces exerting on each masses.

Then I used my two Newton's laws, to find out the relation between:

1) The tension generated on each masses (Newton third law)
2) Acceleration of M2 based on the tension generated by M2

Here is my approach:

I wrote down:

##\sum_{}^{} \vec{F}_{M2} = m_{M2}\vec{a}_{M2}##
##\sum_{}^{} \vec{F}_{M1} = m_{M1}\vec{a}_{M1}##

So, with ##\vec{T}## being the tension applied to the string, and ##\vec{R}## the resistance from the table:

##\vec{T}_{M2} - \vec{W}_{M2} = m_{M2}\vec{a}_{M2}##
##\vec{R}_{M1} - \vec{W}_{M1} + \vec{T}_{M1} = m_{M1}\vec{a}_{M1}##

I simplified the equation for M1, by suppressing ##\vec{R}_{M1}## and ##\vec{W}_{M1}## as ##\vec{R}_{M1} - \vec{W}_{M1} = 0##
I concluded that tension generated by M2 was the tension applied on M1

So,
##\vec{T}_{M2} = - \vec{T}_{M1}##
##\vec{T}_{M1} = - m_{M2}\vec{g}##

using that relation, we can write:

## \vec{a}_{M1} = \frac{m_{M2}}{m{_M1}}g##

Differentiating , I found that:

##\vec{v}(t) = v_{0} + \frac{m_{M2}}{m{_M1}}gt ## with ## v_{0} = 0##
##\vec{x}(t) = x_{0} + \frac{m_{M2}}{2m{_M1}}gt^2 ## with ## x_{0} = 0##

As per the clue given in the manual, I should have a "4" instead of a "2" on the last equation... Looks like my mistake is coming for my differentiation, but I have no idea why... The "4" will come when differentiating a ##x^3##.

Don't give me the answer, I just want to have clues as I'm looking to improve (I'm not involved in any kind of scholarship at this time, and I'm doing that for my personal pleasure :D).

Sorry if my English is not the best, English is not my first language (but I'm trying to improve as well)

Thanks
Have a good day
Rémi
 

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  • #2
It appears that you have assumed that the force ##\vec{T_{M2}}## acting on ##M_2## from tension in the string is given by ##g\ m_{M_2}##.

How did you arrive at that?
 
  • #3
Hello,

You are right ! I just assumed that the magnitude of ##\vec{T_{M2}}## and ##\vec{W_{M2}}## were equal.
But, now I realize that it's a nonsense as we have an accelerated movement towards the ground.

I will start the exercise again, and see how I can figure out ##\vec{T_{M2}}##.
I'll will come back later with the solution

Thanks
Rémi
 
  • Like
Likes jbriggs444

Related to You're welcome, Rémi! Good luck with your exercise.

1. What is the purpose of using two masses and a pulley in an experiment?

The use of two masses and a pulley in an experiment allows for the study of forces and motion. The masses are connected by a string or rope that runs over the pulley, creating a system in which one mass can exert a force on the other through the tension in the string. This allows for the examination of concepts such as acceleration, velocity, and Newton's laws of motion.

2. How does the placement of the pulley affect the results of the experiment?

The placement of the pulley can greatly affect the results of the experiment. If the pulley is placed on a frictionless surface, it will not add any resistance to the system and the results will be more accurate. However, if the pulley is placed on a surface with friction, it can impact the tension in the string and alter the results.

3. What factors can affect the acceleration of the two masses?

The acceleration of the two masses can be affected by several factors, such as the mass of the objects, the angle of the string, and the presence of friction. The larger the mass of the objects, the greater the force needed to accelerate them. A steeper angle of the string will also result in a greater acceleration. Friction can reduce the acceleration by acting as a resistive force.

4. How does the direction of the force impact the motion of the masses?

The direction of the force applied to the masses will determine the direction of their motion. If the force is applied in the same direction as the motion, the acceleration will increase. If the force is applied in the opposite direction, it will act as a resistive force and decrease the acceleration. The direction of the force also affects the tension in the string, which can impact the motion of the masses.

5. Can the masses in a two mass and pulley system move at different speeds?

Yes, the masses in a two mass and pulley system can move at different speeds. The relationship between the masses and the tension in the string determines the acceleration of each mass. If the masses have different masses, they will accelerate at different rates and therefore move at different speeds. This can also be affected by the presence of friction and other factors mentioned previously.

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