# Calculating Length of Arc BD with Hookes Law

• Dell
In summary, we are trying to calculate the change in the length of the arc BD and the new angle DAB using the given stresses and strains. To do this, we can use the transformation equations and the given equations for strain and volume. However, when using this method, we get an incorrect answer.
Dell
in the following question,

E=65 GPa
V=0.3

find the new length of the arc BD??

i have found the stresses

$$\sigma$$xx=-56Mpa
$$\sigma$$yy=0
$$\sigma$$xy=-28Mpa

using hookes law i can find the strains

$$\epsilon$$xx=-8.615e-5
$$\epsilon$$yy=2.58e-4
0.5*$$\epsilon$$xy=$$\gamma$$=-1.12e-3

but how do i calculate the change in the arc using this? i would know how to solve this if i had some kind of angular strain- i need to use a polar system not Cartesian. is there any way to do this?

also how do i know the new angle DAB? i know that the XY axis' new angle is 90.06417, and the n,t system (axes tilted 45 degrees to XY) is also 90.06417 but how do i find DAB,? generally is there any way of knowing how the axis is strained, for example, has the X axis dropped 0.06417 degrees, or the Y axis opened up 0.06417 degrees, or a bit each??
in this specific case can i say that since there is no yy strain the x-axis stays at the same angle?

DA*=DA(1+$$\epsilon$$tt)=4.999569cm
AB*=5.0012923cm

Last edited:
can i do this:

using the transformation equations, i know

εnn= (εxx + εyy)/2 + (εxx - εyy)/2*cos(2ϴ) + εxxsin(2ϴ)

since i have already found xx, yy, xy, instead of looking for a specific εnn can i take the whole eqaution and say

ΔL=$$\int$$εnndL {dL=r*dϴ}

=$$\int$$εnn*r*dϴ with my integral going from 0 to pi/4

is this a possibility?

i tried the following logic,
since in this special specific case, i have found that $$\epsilon$$xx = $$\epsilon$$AD, i know that the radiiii will stay rhe same lengths as each other after deformation therefore preserving the circular shape of the arc

knowing that the volume of the shape with an area of an eighth of a circle (DAB) before deformation is V and after deformation is V'
lets say the thickness of the board is "t"

the new angle DAB is " a' " after deformation

$$\Delta$$=($$\epsilon$$xx +$$\epsilon$$yy + $$\epsilon$$zz)

V=(pi*R2)t/8

V'= (a')(R')2(t')/2

but i also know that

V'=$$\Delta$$*(1+V)
R'=R(1+$$\epsilon$$xx)
t'=t(1+$$\epsilon$$zz)

(a')(R')2(t')=(pi*R2)t/8*(1+$$\epsilon$$xx +$$\epsilon$$yy + $$\epsilon$$zz)

(a')(R(1+$$\epsilon$$xx))2(t(1+$$\epsilon$$zz))=(pi*R2)t/8*(1+$$\epsilon$$xx +$$\epsilon$$yy + $$\epsilon$$zz)therfore i get

(a')= (pi/4)*(1+$$\epsilon$$xx +$$\epsilon$$yy + $$\epsilon$$zz)/[(1+$$\epsilon$$xx))2((1+$$\epsilon$$zz))]

once i have the new angle, since it is still an arc of a circle

L'=R'*a'
L=R*pi/4

$$\delta$$L=L'-L

but this gives me an incorrect answer

is this a correct method and do i maybe have something wrong in y calculations

i get $$\delta$$L=1.016463761198405e-005

## 1. What is Hookes Law and how does it relate to calculating the length of arc BD?

Hookes Law states that the extension or compression of a spring is directly proportional to the applied force, as long as the elastic limit of the spring is not exceeded. This means that the length of arc BD can be calculated by using Hookes Law to determine the amount of force applied to the spring.

## 2. How do you determine the force applied to the spring in order to calculate the length of arc BD?

The force applied to the spring can be determined by measuring the displacement of the spring from its original length and multiplying it by the spring constant, which is a measure of the stiffness of the spring. This force can then be used in Hookes Law to calculate the length of arc BD.

## 3. Is there a specific formula for calculating the length of arc BD with Hookes Law?

Yes, the formula for calculating the length of arc BD with Hookes Law is: arc BD = (force applied x spring constant) / (pi x radius).

## 4. Can Hookes Law be used to calculate the length of arc BD for any type of spring?

Yes, Hookes Law can be used to calculate the length of arc BD for any type of spring as long as the elastic limit is not exceeded. However, the spring constant may vary depending on the type of spring being used.

## 5. What are some factors that can affect the accuracy of calculating the length of arc BD with Hookes Law?

The accuracy of calculating the length of arc BD with Hookes Law can be affected by factors such as the precision of the measurements taken, the condition of the spring, and external forces acting on the spring. It is important to use precise measurements and a well-maintained spring in order to obtain accurate results.

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