Question 256: Properties of Matter

Question

An elastic spring under tension of $$3 \mathrm{~N}$$ has a length $$a$$. Its length is $$b$$ under tension $$2 \mathrm{~N}$$. For its length $$(3 a-2 b)$$, the value of tension will be _______ N.

Accepted Answer

To determine the tension for the elastic spring's length of $$(3a - 2b)$$, we can use Hooke's law which states that the force exerted by a spring is proportional to the extension or compression of the spring from its natural length. Mathematically, Hooke's law is given by:
$$ F = k \cdot \Delta x $$
where:
$$F$$ is the force exerted by the spring.
$$k$$ is the spring constant (a measure of the stiffness of the spring).
$$\Delta x$$ is the displacement from the natural length.
Let's denote the natural (unstretched) length of the spring as $$l_0$$. Given that the spring's length is $$a$$ under a tension of $$3 \mathrm{~N}$$, and its length is $$b$$ under a tension of $$2 \mathrm{~N}$$, we can write the equations as:
$$ 3 \mathrm{~N} = k \cdot (a - l_0) $$
$$ 2 \mathrm{~N} = k \cdot (b - l_0) $$
Next, we need to find the length displacement $$(3a - 2b)$$ in terms of the natural length $$l_0$$. We express the displacements as:
$$ x = (3a - 2b) - l_0 $$
To solve for this, let's express $$a - l_0$$ and $$b - l_0$$ from the given conditions:
$$ a - l_0 = \frac{3 \mathrm{~N}}{k} $$
$$ b - l_0 = \frac{2 \mathrm{~N}}{k} $$
Substitute these into the displacement equation:
$$ x = (3a - 2b) - l_0 = 3 \left( \frac{3 \mathrm{~N}}{k} + l_0 \right) - 2 \left( \frac{2 \mathrm{~N}}{k} + l_0 \right) - l_0 $$
On simplifying, we get:
$$ x = 3 \frac{3 \mathrm{~N}}{k} + 3 l_0 - 2 \frac{2 \mathrm{~N}}{k} - 2 l_0 - l_0 $$
$$ x = \frac{9 \mathrm{~N}}{k} + 3 l_0 - \frac{4 \mathrm{~N}}{k} - 3 l_0 $$
$$ x = \frac{5 \mathrm{~N}}{k} $$
Finally, by Hooke's Law, the force corresponding to this displacement is:
$$ F = k \cdot x = k \cdot \frac{5 \mathrm{~N}}{k} = 5 \mathrm{~N} $$
Therefore, the tension required for the elastic spring's length $$(3a - 2b)$$ is
5 N
.

Properties of Matter

An elastic spring under tension of $3 \mathrm{~N} has a length a. Its length is b under tension 2 \mathrm{~N}. For its length (3 a-2 b)$, the value of tension will be _______ N.

Integer / Numeric Answer Type