If the work done in stretching a wire by 1 mm is 2 J, the work necessary for stretching another wire of the same material but double the radius and half the length by 1 mm is

Explanation

Work Done in Stretching a Wire:The work done in stretching a wire depends on two factors:Force (F): The force applied to stretch the wire.Distance (d): The amount by which the wire is stretched.We can express the work done (W) as the product of force and distance:W = F * dRelationship Between Force and Cross-Sectional Area:For a wire of a given material, the force required to stretch it is proportional to the cross-sectional area of the wire. This is because a wider wire can distribute the stretching force over a larger area, making it more resistant to stretching.Impact of Doubling the Radius:If the radius of the wire is doubled, the cross-sectional area increases by a factor of 4 (since area is proportional to the square of the radius). This means the force required to stretch the wire also increases by a factor of 4.Impact of Halving the Length:If the length of the wire is halved, the distance it needs to be stretched to achieve the same 1 mm elongation is also halved.Combining the Effects:When we combine the effects of doubling the radius and halving the length:The force required to stretch the wire increases by 4.The distance the wire needs to be stretched is halved.Overall Work Done:Since the force increases by 4 and the distance is halved, the overall work done (which is the product of force and distance) remains the same:New work = (4 * original force) * (0.5 * original distance) = original workTherefore, the work done in stretching the second wire (with double the radius and half the length) by 1 mm is also 2 J, which is the same as the work done in stretching the first wire.