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

[tze liu]

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?

 

[andrevdh]

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.

 

[tze liu]

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

 

[andrevdh]

The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.

 

[tze liu]

The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.

The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.

 

[tze liu]

The blocks on the inclines are sharing the horizontal acceleration of the prism since they are moving with it.

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??

 

[andrevdh]

The accelerations of the blocks along the incline are the same, say a , since they are connected with a string.

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

only along the inclines we have a

 

[Ray Vickson]

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.)

 

[andrevdh]

So you could analyze the forces in the horizontal direction.

 

[tze liu]

So you could analyze the forces in the horizontal direction.

Did I do something wrong in these several steps thank

 

[tze liu]

Can u use high school method to do it how about my work done here?

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.)

 

[andrevdh]

The a_1x and the a_2x 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 a_o in all three equations.
This approach I guess would be more promising, also it directly uses the symbols suggested in the original question.

 

[tze liu]

The a_1x and the a_2x 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 a_o in all three equations.
This approach I guess would be more promising, also it directly uses the symbols suggested in the original question.

The a_1x and the a_2x 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 a_o 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

 

[andrevdh]

The original question wants you to find a relationship between a and a_o, that is maybe a/a_o = ... in terms of the masses and the angles and what not

so then you would use the five equations to answer the questions - that is find the simplest relationship.

 

[tze liu]

The original question wants you to find a relationship between a and a_o, that is maybe a/a_o = ... in terms of the masses and the angles and what not

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

 

[rahularvind06]

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.

 

[haruspex]

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?

 

[andrevdh]

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 a_o, 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 a_o. This will give you 2 more equations.

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

That gives you 5 equations in total.

 

[andrevdh]

Your x-equations are correct, say for 2

m₂ g sin(α₂) - T = m₂a

but the block is also accelerating horizontally a_o

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

 

[tze liu]

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

 

[haruspex]

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.

 

[andrevdh]

just try and set up the equation for the horizontal components of the forces on the blocks