Solid mechanics-how do i find the maximum momentum in this question?

In summary, to calculate the maximum momentum in a solid mechanics problem, you need to find the total momentum of the system by multiplying the mass of each object by its velocity and adding these values together. The formula for determining maximum momentum is M = mv, where m is the mass and v is the velocity of the object. The concept of impulse is closely related to maximum momentum and can be used to consider the forces and time applied to the system. The maximum momentum can be affected by factors such as mass, velocity, direction, and external forces. The principle of conservation of momentum can also be used to find the maximum momentum by setting up equations and solving for unknown variables.
  • #1
Dell
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in this question, i am given the function for the xx stress as seen in the diagram below:
http://lh6.ggpht.com/_H4Iz7SmBrbk/SvG3wUaPVAI/AAAAAAAAB6w/sMKXVGWomi8/Capture1.JPG [Broken]

i am asked to sketch the stress distribution, which i have done
http://lh3.ggpht.com/_H4Iz7SmBrbk/SvG3wPYlmdI/AAAAAAAAB6s/rd9T_ED6MK0/Capture.JPG [Broken]

now i am asked to find the value of C which will give me the maximum value for Mo, which as far as i can see is Moment about the z axis.

what I've done is

Mz= -ʃʃ {(σxx)*y} dA

now i know that there are 3 options,(3 functions for σxx) but i know that for y>=c and y<c the function for σxx is constant and -ʃʃ {y} dA is 0,(2nd area moment)

therefore i think i must find it where -c<y<c,

-ʃʃ {(-σy/c)*y} dA= σ/c * ʃʃ {(y^2} dA
=σ/c * ʃʃ {(y^2} dydz [(z from -b/2 to b/2), (y from -h/2 to h/2)]
and i get

Mo=(σ*h^3*b)/(12c)

now how do i find thevalue for c that will give me the maximum Mo?? if c=0 Mo is infinite ??

i thought maybe to find the derivative and compare to 0 but by what? y? z? either way i don't see how that would help, since i am looking for the maximum "c" value, then i thought maybe dMo/dC but i don't think so.
 
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  • #2


Hello,

To find the value of C that will give you the maximum value for Mo, you can use the first derivative test. This means taking the derivative of Mo with respect to C and setting it equal to 0. Then, you can solve for C.

Let's start with the equation you have for Mo:

Mo=(σ*h^3*b)/(12c)

To find the derivative, we can use the quotient rule:

dMo/dC = -(σ*h^3*b)/12c^2

Setting this equal to 0 and solving for C, we get:

0 = -(σ*h^3*b)/12c^2
0 = -(σ*h^3*b)
c^2 = (σ*h^3*b)/12
c = √((σ*h^3*b)/12)

So, the value of C that will give you the maximum value for Mo is √((σ*h^3*b)/12). This makes sense because as C increases, the value of Mo decreases. Therefore, the maximum value for Mo will occur when C is at its minimum, which is √((σ*h^3*b)/12).
 
  • #3


To find the maximum momentum in this question, you will need to use the fundamental principles of solid mechanics, specifically the equations of equilibrium and the equations of motion. The first step would be to set up the equations of equilibrium for the system shown in the diagram. This will involve balancing the external forces and moments acting on the system with the internal stresses and strains.

Next, you will need to use the equations of motion to relate the stresses and strains to the external loads and moments. This will allow you to determine the maximum value of the moment about the z-axis, Mz, for a given value of C.

To find the value of C that will give you the maximum value of Mz, you can use the derivative of Mz with respect to C, dMz/dC, and set it equal to 0. This will give you the critical value of C that will result in the maximum moment.

Alternatively, you can also use the principle of virtual work to determine the value of C that will give you the maximum Mz. This involves taking the virtual moment about the z-axis and setting it equal to 0, and then solving for the value of C.

In summary, to find the maximum momentum in this question, you will need to use the fundamental principles of solid mechanics, specifically the equations of equilibrium and motion, and the principle of virtual work. By setting up and solving these equations, you will be able to determine the value of C that will give you the maximum moment about the z-axis.
 

1. How do I calculate maximum momentum in solid mechanics?

In order to find the maximum momentum in a solid mechanics problem, you will need to first calculate the total momentum of the system. This can be done by multiplying the mass of each object in the system by its velocity, and then adding all of these values together. The maximum momentum will occur when the system is at its highest velocity.

2. What is the formula for determining maximum momentum?

The formula for determining maximum momentum in solid mechanics is: M = mv, where m is the mass of the object and v is the velocity. This formula can be applied to each object in the system to find the total momentum, and then the maximum momentum can be determined.

3. How does the concept of impulse relate to maximum momentum?

The concept of impulse is closely related to maximum momentum in solid mechanics. Impulse is defined as the change in momentum of an object, and is equal to the force applied to the object multiplied by the time it is applied. In order to find the maximum momentum, you will need to consider the impulse applied to the system and how it affects the velocity of the objects.

4. What factors can affect the maximum momentum in a solid mechanics problem?

The maximum momentum in a solid mechanics problem can be affected by a variety of factors. These may include the mass of the objects, the velocity of the objects, the direction of the velocity, and the forces acting on the objects. Additionally, external factors such as friction and air resistance can also impact the maximum momentum.

5. How can I use the concept of conservation of momentum to find the maximum momentum?

The principle of conservation of momentum states that the total momentum of a closed system remains constant. This means that the total momentum before and after any interaction within the system will be equal. In order to find the maximum momentum, you can use this principle to set up equations and solve for the unknown variables, such as velocity or mass.

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