A copper wire sags 54.0 cm between two utility poles 30.0 m apart when the temperature is -15° C. Estimate the amount of sag when the temperature is + 35° C. [Hint: An estimate can be made by assuming the shape of the wire is approximately an arc of a circle; hard equations can sometimes be solved by guessing values.]

Verified step by step guidance

1. Understand the problem: The sag of the copper wire changes with temperature due to thermal expansion. The wire can be approximated as an arc of a circle, and the change in sag is related to the change in the length of the wire caused by thermal expansion.
2. Step 1: Calculate the initial length of the wire at -15°C. The wire spans 30.0 m horizontally, and the sag is 54.0 cm (0.54 m). Using the geometry of a circle, approximate the arc length of the wire. Assume the arc length is slightly longer than the straight-line distance between the poles.
3. Step 2: Use the formula for thermal expansion to calculate the change in length of the wire as the temperature increases from -15°C to +35°C. The formula is ΔL = L₀αΔT, where L₀ is the initial length of the wire, α is the coefficient of linear expansion for copper (approximately 1.7 × 10⁻⁵ /°C), and ΔT is the temperature change.
4. Step 3: Add the change in length (ΔL) to the initial length (L₀) to find the new length of the wire at +35°C. This new length will determine the new sag of the wire.
5. Step 4: Approximate the new sag by considering the geometry of the wire as an arc of a circle. The increased length of the wire will result in a larger sag. Use trial and error or an iterative approach to estimate the new sag, ensuring the arc length matches the new length of the wire.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of materials to change their dimensions in response to temperature changes. In metals like copper, an increase in temperature causes the atoms to vibrate more, leading to an increase in length. This concept is crucial for understanding how the wire will stretch and sag differently at varying temperatures.

Catenary Curve

A catenary curve describes the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force. In this problem, the sagging wire can be approximated as a segment of a catenary, which helps in estimating the change in sag due to temperature variations. Understanding this shape is essential for visualizing how the wire behaves under different conditions.

Mechanical Properties of Materials

The mechanical properties of materials, such as tensile strength and elasticity, determine how a material deforms under stress. For copper wire, these properties influence how much it will sag when subjected to its own weight and temperature changes. Recognizing these properties is vital for predicting the wire's behavior as temperature fluctuates.