The problem involves two wires,
A and
B, made from the same material. The ratio of their diameters is
1:2, and the ratio of their lengths is
1:3. When the same force is applied to both wires, we need to determine the ratio of their length increases.
Here's how we can solve it:
The formula for the increase in length (
∆ℓ ) when a wire is stretched by a force
F is given by:
∆ℓ== Where:
ℓ is the original length,
A is the cross-sectional area of the wire,
Y is Young's modulus,
r is the radius of the wire.
To find the ratio of the increase in lengths of wires
A and
B, we calculate:
=()×()2For wires A and B :
The length ratio
ℓ1:ℓ2 is
1:3.
The diameter ratio (
⇒ radius ratio) is
1:2, so
=.
Substitute these values into the equation:
=×()2=×4=Therefore, the ratio of the increase in their lengths is
4:3.