In Fig. 10.20, the steel ring of diameter 2.5 cm is 0.10 mm smaller in diameter than the steel ball at 20° C. (a) For the ball to go through the ring, should you heat (1) the ring, (2) the ball, or (3) both? Why? (b) What is the minimum required temperature?
To answer both parts of the question, we need to use the concept of thermal expansion. When an object is heated, it expands due to the increase in its temperature. In this case, we have a steel ring and a steel ball.
(a) To find out whether we should heat the ring, the ball, or both, we need to understand the effect of heating on their sizes. When an object expands, its dimensions increase, and when an object contracts, its dimensions decrease.
Given that the ring is smaller in diameter than the ball, we need to make the ring larger in order for the ball to go through it. Therefore, we need to heat the ring. By heating the ring, it will expand, increasing its diameter, and creating enough space for the ball to pass through.
(b) Now, let's determine the minimum required temperature. To calculate this, we'll use the coefficient of linear expansion (α) for steel. The coefficient of linear expansion represents the amount of expansion or contraction of a material per degree Celsius (or Kelvin) change in temperature.
To find the coefficient of linear expansion for steel, we can refer to reference materials or use the commonly accepted value of α = 12 x 10^(-6) °C^(-1) (varies slightly depending on the type of steel).
Given that the ring is 0.10 mm smaller in diameter than the ball, we can convert this difference to linear expansion. Since the diameter is twice the radius, the difference in radius is 0.10 mm / 2 = 0.05 mm = 0.05 x 10^(-3) m.
Using the formula for linear expansion, ΔL = α * L * ΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L is the initial length, and ΔT is the change in temperature, we can rearrange it to find the change in temperature:
ΔT = ΔL / (α * L)
To find the change in temperature required to expand the ring by 0.05 x 10^(-3) m, we need to know the initial length of the ring. However, only the diameter is provided, which is 2.5 cm. To find the initial length, we'll use the formula: L = π * d, where L is the length and d is the diameter.
Plugging in the values, L = π * 2.5 cm = π * 2.5 x 10^(-2) m.
Now, we can calculate the change in temperature:
ΔT = (0.05 x 10^(-3) m) / (12 x 10^(-6) °C^(-1) * π * 2.5 x 10^(-2) m)
Simplifying the equation, we get:
ΔT = (0.05 x 10^(-3) m) / (12 x 10^(-6) °C^(-1) * 3.14 * 2.5 x 10^(-2) m)
ΔT ≈ 0.00505 °C
Hence, the minimum required temperature for the ring to expand enough for the ball to pass through is approximately 0.00505 °C.