2. A certain wire of length L, cross-sectional area A is extended by an amount ‘e’
by a force F. Given that the young’s modulus of the material of the wire is E,
Show that the;
a. Energy stored in the wire is given by the expression (5)
b. Energy per
unit volume, Ev = x (strain) 2 (5)
3. A tensile force of 180 N when applied to a steel wire of length 2 m and cross
sectional area 0.3 mm 2 extends the wire by 0.6 cm. calculate
a. Young’s modulus of the elasticity of the wire. (3)
b. The energy stored in the wir e.
a. To show that the energy stored in the wire is given by the expression (5), we need to use the formula for the strain energy stored in a wire under tension. The strain energy, U, is given by the equation:
U = 1/2 * stress * strain * volume
In this case, the volume of the wire can be expressed as the product of the cross-sectional area, A, and the extension, e:
V = A * e
The stress can be calculated as the force, F, divided by the cross-sectional area:
stress = F / A
And the strain is the extension, e, divided by the original length, L:
strain = e / L
Substituting these values into the strain energy equation, we get:
U = 1/2 * (F / A) * (e / L) * (A * e)
Simplifying the expression, we get:
U = 1/2 * F * e^2 / L
This is the expression (5) for the energy stored in the wire.
b. To show that the energy per unit volume, Ev, is given by the expression (5), we need to express the energy stored, U, in terms of the volume, V, instead of the length, L. Using the same strain energy equation as before, we substitute V = A * e:
U = 1/2 * F * e^2 / L
And since V = A * e, we can express e in terms of V and A:
e = V / A
Substituting this expression back into the energy equation, we get:
U = 1/2 * F * (V / A)^2 / L
The energy per unit volume, Ev, is then given by:
Ev = U / V = 1/2 * F * (V / A)^2 / (L * V) = 1/2 * F * (V / A / L)
Simplifying further, we get:
Ev = 1/2 * F / (A * L)
This is the expression (5) for the energy per unit volume.
Now let's move on to question 3.
a. To calculate the Young's modulus of elasticity, we can use the formula:
E = stress / strain
Given that the tensile force, F, is 180 N and the extension, e, is 0.6 cm (which is equivalent to 0.006 m), and the original length, L, is 2 m, and the cross-sectional area, A, is 0.3 mm^2 (which is equivalent to 0.3 * 10^(-6) m^2), we can calculate the stress and strain:
stress = F / A = 180 N / 0.3 * 10^(-6) m^2 = 600 * 10^6 N/m^2
strain = e / L = 0.006 m / 2 m = 0.003
Now we can calculate the Young's modulus of elasticity:
E = stress / strain = 600 * 10^6 N/m^2 / 0.003 = 200 * 10^9 N/m^2
Therefore, the Young's modulus of elasticity of the wire is 200 * 10^9 N/m^2.
b. To calculate the energy stored in the wire, we can use the formula for strain energy:
U = 1/2 * stress * strain * volume
Given that the cross-sectional area, A, is 0.3 mm^2 (which is equivalent to 0.3 * 10^(-6) m^2), the extension, e, is 0.6 cm (which is equivalent to 0.006 m), the tensile force, F, is 180 N, and the original length, L, is 2 m, we can calculate the volume:
V = A * e = 0.3 * 10^(-6) m^2 * 0.006 m = 1.8 * 10^(-9) m^3
Now we can calculate the energy stored:
U = 1/2 * stress * strain * volume = 1/2 * 600 * 10^6 N/m^2 * 0.003 * 1.8 * 10^(-9) m^3 = 0.54 * 10^(-3) J
Therefore, the energy stored in the wire is 0.54 * 10^(-3) J.
a) To derive the expression for the energy stored in the wire, we can start by considering the relationship between the force applied, the extension of the wire, and the Young's modulus.
According to Hooke's Law, the force F applied to a wire is directly proportional to the extension e, and the constant of proportionality is the Young's modulus (E):
F = E * (e/L) (1)
where L is the original length of the wire.
The work done in stretching a wire is given by the formula:
W = F * e (2)
By substituting the value of F from equation (1) into equation (2), we get:
W = E * (e/L) * e
W = (E * e^2) / L
The energy stored in the wire (U) is equal to the work done, which is given by:
U = W
Therefore,
U = (E * e^2) / L
This is the expression for the energy stored in the wire.
b) The energy per unit volume (Ev) is defined as the energy stored in the wire per unit volume of the wire.
The cross-sectional area of the wire is A, and the volume of a wire can be expressed as:
V = A * L (3)
where L is the length of the wire.
To find the energy per unit volume, we divide the energy stored (U) by the volume (V) of the wire:
Ev = U / V
Substituting the values of U from equation (a) and V from equation (3), we get:
Ev = (E * e^2) / (A * L)
Therefore, the expression for the energy per unit volume is:
Ev = (E * e^2) / (A * L) x (strain)^2
This is the expression for the energy per unit volume in terms of the strain.
Now let's move on to question 3.
3) a) To calculate Young's modulus (E) of the wire, we can use the formula derived in part a:
F = E * (e/L)
Rearranging the formula, we can solve for E:
E = F * (L/e)
Given that the force (F) is 180 N, the length of the wire (L) is 2 m, and the extension (e) is 0.6 cm = 0.006 m, we can substitute these values into the equation:
E = 180 N * (2 m / 0.006 m)
Simplifying this equation will give us the value of Young's modulus.
b) To calculate the energy stored in the wire, we can use the formula derived in part a:
U = (E * e^2) / L
Given the values of E, e, and L, we can substitute them into the formula to calculate the energy stored in the wire.