A Newton's law question which is too hard for me

In summary, the problem involves a triangular prism on a frictionless horizontal plane with two inclined sides and two blocks connected by a string sliding on the prism. The goal is to find the acceleration of the blocks relative to the prism and the ratio of masses for the prism to be in equilibrium. This can be solved by using the Lagrangian form of mechanics with generalized coordinates and analyzing the forces in both the horizontal and inclined directions. The horizontal forces involve an acceleration ao for all three objects, while the inclined forces involve a common acceleration a for the blocks.
  • #1
tze liu
57
0

Homework Statement



Question 1.
A triangular prism of mass M is placed one side on a frictionless horizontal plane as shown in Fig. 1. The other two sides are inclined with respect to the plane at angles a1 and
a2 respectively. Two blocks of masses m1 and m2, connected by an inextensible thread, can slide without friction on the surface of the prism. The mass of the pulley, which supports the thread, is negligible.

  • Express the acceleration a of the blocks relative to the prism in terms of the acceleration a0 of the prism.
  • Find the acceleration a0 of the prism in terms of quantities given and the acceleration g due to gravity.
  • At what ratio m1/m2 the prism will be in equilibrium?

Homework Equations


ma=F

The Attempt at a Solution


see the picture below

i got totally wrong as it is too hard
can everyone give me hint to do this question
How did i do this question in a more clear and systematic way?
 

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  • #2
I think you assumed that the blocks are not accellerating in the directions perpendicular to the inclined surfaces of the prism, which of cause not true.
 
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  • #3
andrevdh said:
I think you assumed that the blocks are not accellerating in the directions perpendicular to the inclined surfaces of the prism, which of cause not true.
But how to do this question thank
 
  • #4
The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.
 
  • #5
andrevdh said:
The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.
andrevdh said:
The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.
 

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  • #6
andrevdh said:
The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.
andrevdh said:
The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.
seems i forget to use consider the inertial force in this case??
 
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  • #7
The accelerations of the blocks along the incline are the same, say a , since they are connected with a string.

blocks sliding on prism inclines.jpg


They are also accelerating horizontally, ao , so I am not convinced it is business like usual :sorry:

Picture 24.jpg


only along the inclines we have a
 
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  • #8
tze liu said:

Homework Statement



Question 1.
A triangular prism of mass M is placed one side on a frictionless horizontal plane as shown in Fig. 1. The other two sides are inclined with respect to the plane at angles a1 and
a2 respectively. Two blocks of masses m1 and m2, connected by an inextensible thread, can slide without friction on the surface of the prism. The mass of the pulley, which supports the thread, is negligible.

  • Express the acceleration a of the blocks relative to the prism in terms of the acceleration a0 of the prism.
  • Find the acceleration a0 of the prism in terms of quantities given and the acceleration g due to gravity.
  • At what ratio m1/m2 the prism will be in equilibrium?

Homework Equations


ma=F

The Attempt at a Solution


see the picture below

i got totally wrong as it is too hard
can everyone give me hint to do this question
How did i do this question in a more clear and systematic way?

Have you taken the Lagrangian form of Mechanics yet? If so, the problem is a relatively straightforward application of the Lagrangian, with "generalized coordinates" ##x## = horizontal location of the peak of the prism, and ##u##= length of the pulley-##m_1## string. The length of the pulley-##m_2## string is ##T-u##, where ##T## = total length of the string (fixed). If ##H## is the "height" of the prism, the (x,y) coordinates of mass ##m_1## are ##x_1 = x - u \cos(\alpha_1)## and ##y_1 = H - u \sin(\alpha_1)##. The (x,y) coordinates of mass ##m_2## are ##x_2 = x + (T-u) \cos(\alpha_2)## and ##y_2 = H - (T-u) \sin(\alpha_2)##. From that we can get the velocity vectors ##(\dot{x},0)## of the prism and ##(\dot{x}_1, \dot{y}_2), (\dot{x}_2, \dot{y}_2)## of masses ##m_1, n_2##, and can thus work out their kinetic energies, expressing everything in terms of ##x, u, \dot{x}, \dot{u}##.. We can also get the potential energies of masses ##m_1,m_2##. (The prism has potential energy as well, but it remains constant because the prism moves horizontally.)
 
  • #9
So you could analyze the forces in the horizontal direction.
 
  • #10
andrevdh said:
So you could analyze the forces in the horizontal direction.
Did I do something wrong in these several steps thank
 

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  • #11
Can u use high school method to do it how about my work done here?
Ray Vickson said:
Have you taken the Lagrangian form of Mechanics yet? If so, the problem is a relatively straightforward application of the Lagrangian, with "generalized coordinates" ##x## = horizontal location of the peak of the prism, and ##u##= length of the pulley-##m_1## string. The length of the pulley-##m_2## string is ##T-u##, where ##T## = total length of the string (fixed). If ##H## is the "height" of the prism, the (x,y) coordinates of mass ##m_1## are ##x_1 = x - u \cos(\alpha_1)## and ##y_1 = H - u \sin(\alpha_1)##. The (x,y) coordinates of mass ##m_2## are ##x_2 = x + (T-u) \cos(\alpha_2)## and ##y_2 = H - (T-u) \sin(\alpha_2)##. From that we can get the velocity vectors ##(\dot{x},0)## of the prism and ##(\dot{x}_1, \dot{y}_2), (\dot{x}_2, \dot{y}_2)## of masses ##m_1, n_2##, and can thus work out their kinetic energies, expressing everything in terms of ##x, u, \dot{x}, \dot{u}##.. We can also get the potential energies of masses ##m_1,m_2##. (The prism has potential energy as well, but it remains constant because the prism moves horizontally.)
 
  • #12
The a1x and the a2x are the same for starters, but I am not convinced this way of approaching it will work (or is the easiest), although I might be wrong.
What i would rather suggest is analyzing the forces along the inclines, as you did, and use just a symbol a for the acceleration along the inclines for both masses, that is your x-direction equations.

The other sets of equations is for all three objects in the horizontal direction, with an acceleration ao in all three equations.
This approach I guess would be more promising, also it directly uses the symbols suggested in the original question.
 
  • #13
andrevdh said:
The a1x and the a2x are the same for starters, but I am not convinced this way of approaching it will work (or is the easiest), although I might be wrong.
What i would rather suggest is analyzing the forces along the inclines, as you did, and use just a symbol a for the acceleration along the inclines for both masses, that is your x-direction equations.

The other sets of equations is for all three objects in the horizontal direction, with an acceleration ao in all three equations.
This approach I guess would be more promising, also it directly uses the symbols suggested in the original question.
andrevdh said:
The a1x and the a2x are the same for starters, but I am not convinced this way of approaching it will work (or is the easiest), although I might be wrong.
What i would rather suggest is analyzing the forces along the inclines, as you did, and use just a symbol a for the acceleration along the inclines for both masses, that is your x-direction equations.

The other sets of equations is for all three objects in the horizontal direction, with an acceleration ao in all three equations.
This approach I guess would be more promising, also it directly uses the symbols suggested in the original question.
then what will you do next!?

thank you
 
  • #14
The original question wants you to find a relationship between a and ao, that is maybe a/ao = ... in terms of the masses and the angles and what not :smile:

so then you would use the five equations to answer the questions - that is find the simplest relationship.
 
  • #15
andrevdh said:
The original question wants you to find a relationship between a and ao, that is maybe a/ao = ... in terms of the masses and the angles and what not :smile:

so then you would use the five equations to answer the questions - that is find the simplest relationship.
i cannot solve it

no idea

as i expand such equations ,it seems that it make it too complicated
 
  • #16
Consider which forces are causing the prism to move, and which forces are causing the blocks to move. Remember the blocks move with the resultant acceleration of a and a0.
 
  • #17
tze liu said:
i cannot solve it

no idea

as i expand such equations ,it seems that it make it too complicated
Please do not post handwritten working as images. Not only are they usually quite hard to read, and frequently sideways or upside down, it also makes it hard to comment on individual steps. Images are for diagrams and textbook extracts.
Suppose the left hand block is accelerating down the slope at rate a relative to the prism. What is the horizontal component of that? If the prism is accelerating to the right at rate a0, what is the net horizontal acceleration of the block?
What net horizontal force must be acting on the block?
Do the same for the right-hand block. What force balance equation does that allow you to write?
 
  • #18
The acceleration of both blocks along the incline is a, say 1 up and 2 down. So develop such equations just like before, just with the acceleration indicated as a for both.

Then for each block you also resolve the forces horizontally. The acceleration of the blocks is ao, let's say horizontally to the right, that is while the blocks are sliding they are moving together with the prism, all accelerating to the right with acceleration ao. This will give you 2 more equations.

Finally one more equation for the prism, since it is also accelerating horizontally to the right at ao.

That gives you 5 equations in total.
 
  • #19
Your x-equations are correct, say for 2

m2 g sin(α2) - T = m2a

but the block is also accelerating horizontally ao

so add up the horizontal components of the forces on the block and they should produce the ao acceleration

block on incline.jpg
 
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  • #20
haruspex said:
Please do not post handwritten working as images. Not only are they usually quite hard to read, and frequently sideways or upside down, it also makes it hard to comment on individual steps. Images are for diagrams and textbook extracts.
Suppose the left hand block is accelerating down the slope at rate a relative to the prism. What is the horizontal component of that? If the prism is accelerating to the right at rate a0, what is the net horizontal acceleration of the block?
What net horizontal force must be acting on the block?
Do the same for the right-hand block. What force balance equation does that allow you to write?
Sorry

i don't know how to post the working more clearly
 
  • #21
tze liu said:
Sorry

i don't know how to post the working more clearly
You can post the diagrams as images as long as you make them large, clear, annotated, and the right way up. There are drawing packages you can get for your computer that will create e.g. .jpg which you can upload. See the image in post #19.
For the algebra, just type it in! Be careful to use parentheses correctly. You can either use the tool icons just above the typing area for superscript (X2), subscript (X2) and special symbols (∑), or learn to use LaTeX. There's a LaTeX help button at the bottom left of the typing area.
 
  • #22
just try and set up the equation for the horizontal components of the forces on the blocks
 

Related to A Newton's law question which is too hard for me

1. What are Newton's Laws of Motion?

Newton's Laws of Motion are a set of three physical laws that describe the relationship between the motion of an object and the forces acting upon it. The laws were developed by Sir Isaac Newton in the 17th century and are considered fundamental principles of classical mechanics.

2. What is the first law of motion?

The first law of motion, also known as the law of inertia, states that an object at rest will remain at rest and an object in motion will remain in motion at a constant velocity unless acted upon by an external force.

3. What is the second law of motion?

The second law of motion states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass. This can be represented by the equation F=ma, where F is the net force, m is the mass, and a is the acceleration.

4. What is the third law of motion?

The third law of motion, also known as the law of action and reaction, states that for every action, there is an equal and opposite reaction. This means that when one object exerts a force on another object, the second object will exert an equal and opposite force back on the first object.

5. How are Newton's laws used in everyday life?

Newton's laws are used in many aspects of everyday life, including driving a car, riding a bike, and playing sports. For example, the first law explains why a car continues to move forward even when the engine is turned off, while the second law is used in designing cars and bicycles to ensure they have enough power to move with the desired acceleration. The third law is also evident in sports such as swimming, where the force of the swimmer's arms pushing back on the water propels them forward.

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