Question

a brass ring with inner diameter 2.00 cm and outer diameter 3.00 cm needs to fit over a 2.00-cm-diameter steel rod, but at 20 c the hole through the brass ring is 50 mm too small in diameter. to what temperature, in c, must the rod and ring be heated so that the ring just barely slips over the rod?

Answer 1

To fit a brass ring over a steel rod, the ring must be heated to expand. Using the formula for thermal expansion and the properties of brass, we calculate the necessary temperature increase to make the ring fit over the rod.

Step-by-step explanation:

Understanding Thermal Expansion for Fitting a Brass Ring Over a Steel Rod

The problem presented deals with the concept of thermal expansion, which is a principle in physics where materials expand when heated. Since the brass ring's hole is 50 mm too small to fit over the steel rod's 2.00 cm diameter at 20°C, we need to calculate the temperature at which the brass will expand enough to slip over the rod. To solve this, we'll use the formula for linear expansion, ΔL = α × L0 × ΔT, where:

• ΔL is the change in diameter
• α is the linear expansion coefficient of brass (typically around 19 × 10^{-6} /°C)
• L0 is the original diameter of the ring's hole
• ΔT is the change in temperature

Since the change in diameter needed is 50 mm or 0.05 cm, and the original diameter is 2 cm (20 mm), we can rearrange the formula to solve for ΔT:

ΔT = ΔL / (α × L0)

ΔT = 0.05 cm / (19 × 10^{-6} /°C × 2 cm)

Upon calculating, we find the temperature change required, which we add to the original temperature of 20°C to find the final temperature needed for the fit. In cases where the coefficient for brass is given differently, use that specific value in calculations.

Answer 2

To make the brass ring fit over the steel rod, the temperature of both the ring and rod needs to be heated to 2500 °C. This is determined using the equation for thermal expansion.

Step-by-step explanation:

To calculate the temperature at which the brass ring will just barely slip over the steel rod, we need to use the equation for thermal expansion. The change in diameter of an object due to temperature change can be given by the equation:

Δd = αd₀ΔT,

where Δd is the change in diameter, α is the coefficient of linear expansion, d₀ is the original diameter, and ΔT is the change in temperature.

In this case, the change in diameter is given as 50 mm (0.05 cm), the original diameter of the brass ring is 2.00 cm, and the original diameter of the steel rod is 2.00 cm. We need to find the change in temperature, ΔT, that will cause the change in diameter. Rearranging the equation, we have:

ΔT = Δd / (αd₀)

Substituting the given values, we have:

ΔT = (0.05 cm) / (2.0 cm × 10⁻⁵ cm/°C) = 2500 °C

Therefore, the temperature at which the ring and rod must be heated is 2500 °C in order for the ring to just barely slip over the rod.