Question

8. At temperature 15°C, aluminum rivets have a diameter of 0.501 cm, and holes drilled in a titanium sheet have a diameter of 0.500 cm. If both the aluminum rivets and the titanium sheet are cooled together, at what temperature will the rivets just fit into the appropriate holes in the titanium sheet? Use 25x10-6 (°C)-1 for the coefficient of linear expansion for aluminum, and 8.5x10-6 (°C)-1 for titanium

Answer 1

The temperature is
`T = -106 ^oC`

Step-by-step explanation:

From the question we are told that

The temperature is
`T_1 = T_t= T_a=15^oC`

The diameter is
`d_1 = 0.5001 cm`

The diameter of the hole
`d_2 = 0.500 \ cm`

The coefficient of linear expansion for aluminum is
`\alpha _1 = 25 *10^(-6) \ ^oC^(-1)`

The coefficient of linear expansion for titanium is
`\alpha _2 = 8.5 *10^(-6) \ ^o C^(-1)`

According to the law of linear expansion

`d = d_o (1 + \alpha \Delta T )`

Where
`d_o` represents the original diameter

So for aluminum

`d_a = d_1 (1 + \alpha_1 (T- T_a) )`

Where
`d_a` is the new diameter of aluminum

`T_a` is the new temperature of the aluminum

So for titanium

`d_t = d_2 (1 + \alpha_1 (T- T_t) )`

Where
`d_t` is the new diameter of titanium

`T_t` is the new temperature of the aluminum

So for the aluminum rivets to fit into the holes

`d_a = d_t`

=>
`d_1 (1 + \alpha_1 (T- T_a) ) = d_2 (1 + \alpha_2 (T- T_t) )`

Making T the subject of the formula

`T = ((d_1 - d_2 ) + (d_2 *\alpha_2 T_t) - d_1 \alpha_1 * T_a )/(d_2 \alpha_2 - d_1 \alpha_1 )`

Substituting values

`T = ((0.501 - 0.500 ) + (0.500 *(8.5*10^(-6)) * 15) - 0.500* (25*10^(-6)) * 15 )/(0.500 * (8.5 *10^(-6)) - 0.501 * (25 *10^(-6)) )`

`T = -106 ^oC`