a metallic ring of radius 'r' and cross sectional area is fitted into an wooden circular disc of radius R (R greater than r).If he young's modulus of the material of the ring is Y, the fore with which the metal ring expands is :
ans:
1.AY(R-r)/r
To find the force with which the metal ring expands, we can use Hooke's Law, which states that the force applied to a material is directly proportional to the strain (change in length) of the material.
In this case, the strain is the change in radius of the ring when it expands. We can calculate this change in radius by considering the change in the cross-sectional area of the ring.
The initial cross-sectional area of the ring is given by A = πr^2, and the final cross-sectional area is A' = πR^2 (since the ring expands and fills the wooden disc).
The change in cross-sectional area is ΔA = A' - A = π(R^2 - r^2).
Now, according to Hooke's Law, the strain (change in length) is given by ε = ΔL / L, where L is the original length.
Since we are dealing with a ring, the original length is the circumference of the ring, which is 2πr.
The change in length or strain is then ε = ΔL / (2πr), where ΔL is the change in circumference.
The change in circumference is ΔL = 2π(R - r).
Substituting the values into the equation, we have ε = (2π(R - r)) / (2πr) = (R - r) / r.
Now, the force applied to the ring is given by F = Y * A * ε, where Y is the Young's modulus.
Substituting the values, we have F = Y * πr^2 * (R - r) / r.
Simplifying further, we get F = Y * π(R - r)r.
Therefore, the force with which the metal ring expands is AY(R - r) / r.
To determine the force with which the metal ring expands, we can use the formula for axial stress:
Stress = (Force/Area) = (Young's Modulus * Strain)
First, let's calculate the strain:
Strain = (Change in length) / (Original length)
Since the ring is expanding, the change in length can be calculated as the difference between the final and initial radii:
Change in length = (R - r)
The original length of the ring is given by 2πr, as it is a circular ring.
So, the strain is:
Strain = (Change in length) / (Original length) = (R - r) / (2πr)
Now, we can substitute the strain into the stress equation:
Stress = (Young's Modulus * Strain)
To find the force with which the ring expands, we need to multiply the stress by the cross-sectional area of the ring. The cross-sectional area of the ring can be calculated as the difference between the areas of two circles:
Cross-sectional area = π(R^2 - r^2)
Now, let's substitute the values:
Force = Stress * Cross-sectional area
= (Young's Modulus * Strain) * π(R^2 - r^2)
= Y * ((R - r) / (2πr)) * π(R^2 - r^2)
= Y * (R - r)(R + r) / (2r)
Therefore, the force with which the metal ring expands is AY(R - r) / r, where A represents the term (R + r) / 2.