A 3000N force stretches a wire by 2.0mm. A second wire of the same material is twice as long and has twice the diameter. How much force is needed to stretch it by 1.0 mm?
To find out how much force is needed to stretch the second wire by 1.0 mm, we can use Hooke's Law, which states that the force required to stretch or compress a spring or wire is directly proportional to the displacement.
Hooke's Law is formulated as follows:
F = k * ΔL
where F is the force applied, k is the spring constant (a measure of the stiffness of the wire), and ΔL is the change in length.
In this case, we have the following information:
For the first wire:
Force (F1) = 3000 N
Change in length (ΔL1) = 2.0 mm
For the second wire:
Change in length (ΔL2) = 1.0 mm
We need to determine the force (F2) required to stretch the second wire.
First, we need to find the spring constant (k) for the first wire:
F1 = k * ΔL1
3000 N = k * 2.0 mm
To find k, we rearrange the equation:
k = F1 / ΔL1
k = 3000 N / 2.0 mm
Now, we can use the spring constant to find the force (F2) required to stretch the second wire:
F2 = k * ΔL2
F2 = (3000 N / 2.0 mm) * 1.0 mm
Let's plug in the values and calculate the force:
F2 = 3000 N / 2.0
F2 = 1500 N
Therefore, the force needed to stretch the second wire by 1.0 mm is 1500 N.
To solve this problem, we can apply Hooke's Law, which states that the force applied to stretch or compress an object is directly proportional to its deformation (change in length), provided the object remains within its elastic limit.
Let's start by calculating the spring constant (k) for the first wire. The spring constant represents the stiffness of the wire and is given by the formula:
k = F / Δx
where:
k = spring constant (N/m)
F = force applied (N)
Δx = change in length or deformation (m)
Given that the force (F) applied to the first wire is 3000N, and the change in length (Δx) is 2.0mm (or 0.002m), we can calculate the spring constant (k1) as follows:
k1 = 3000 N / 0.002 m = 1,500,000 N/m
Now, let's calculate the cross-sectional area (A) of the first wire:
A = π * r^2
where:
A = cross-sectional area (m^2)
π = Pi (approximately 3.14159)
r = radius of the wire (m)
Since the second wire has twice the diameter, its radius (r2) will be twice the radius of the first wire (r1). Therefore, r2 = 2 * r1.
The cross-sectional area (A2) of the second wire can be calculated as follows:
A2 = π * (r2)^2 = π * (2 * r1)^2 = 4 * π * r1^2 = 4 * A1
Next, let's calculate the spring constant (k2) for the second wire using the relationship between the spring constant and the cross-sectional area:
k2 = (F2 * L2) / Δx2 = (F2 * 2 * L1) / Δx2
where:
k2 = spring constant for the second wire (N/m)
F2 = force applied to the second wire (unknown)
L1 = length of the first wire (given)
Δx2 = change in length or deformation of the second wire (0.001m)
Since the first wire is twice as long as the second wire, we have L2 = 2 * L1.
Now, we can compare the two spring constants (k1 and k2) using the relationship:
k1 / k2 = A1 / A2
Substituting the values we've calculated:
(1500000 N/m) / (k2) = A1 / (4 * A1)
Simplifying the equation:
k2 = (4 * A1 * 1500000 N/m) / A1
k2 = 6000000 N/m
Finally, we can use the spring constant (k2) to calculate the force needed to stretch the second wire by 1.0mm (or 0.001m):
F2 = k2 * Δx2
= 6000000 N/m * 0.001m
= 6000N
Therefore, to stretch the second wire by 1.0mm, a force of 6000N is needed.