A cylinder of diameter 1.00000 cm at 30^∘ C is to be slid...

A cylinder of diameter $1.00000 \mathrm{~cm}$ at $30^{\circ} \mathrm{C}$ is to be slid into a hole in a steel plate. The hole has a diameter of $0.99970 \mathrm{~cm}$ at $30^{\circ} \mathrm{C}$. To what temperature must the plate be heated? For steel, $\alpha=1.1 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$ The plate will expand in the same way whether or not there is a hole in it. Hence, the hole expands in the same way a circle of steel filling it would expand. We want the diameter of the hole to change by $$ \Delta L=(1.00000-0.99970) \mathrm{cm}=0.00030 \mathrm{~cm} $$ Using $\Delta L=\alpha L_{0} \Delta T$ $$ \Delta T=\frac{\Delta L}{\alpha L_{0}}=\frac{0.00030 \mathrm{~cm}}{\left(1.1 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)(0.99970 \mathrm{~cm})}=27^{\circ} \mathrm{C} $$ The temperature of the plate must be $30+27=57^{\circ} \mathrm{C}$

A cylinder of diameter $1.00000 \mathrm{~cm}$ at $30^{\circ} \mathrm{C}$ is to be slid into a hole in a steel plate. The hole has a diameter of $0.99970 \mathrm{~cm}$ at $30^{\circ} \mathrm{C}$. To what temperature must the plate be heated? For steel, $\alpha=1.1 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$
The plate will expand in the same way whether or not there is a hole in it. Hence, the hole expands in the same way a circle of steel filling it would expand. We want the diameter of the hole to change by
$$
\Delta L=(1.00000-0.99970) \mathrm{cm}=0.00030 \mathrm{~cm}
$$
Using $\Delta L=\alpha L_{0} \Delta T$
$$
\Delta T=\frac{\Delta L}{\alpha L_{0}}=\frac{0.00030 \mathrm{~cm}}{\left(1.1 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\right)(0.99970 \mathrm{~cm})}=27^{\circ} \mathrm{C}
$$
The temperature of the plate must be $30+27=57^{\circ} \mathrm{C}$

Key Concepts

Thermal Expansion

Thermal expansion is the tendency of matter to change in shape, area, and volume in response to a change in temperature. This phenomenon occurs because materials expand when heated and contract when cooled, and it plays a critical role in engineering and design by affecting how components fit together under different temperatures.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific constant that quantifies how much a material's length changes per unit length for a one-degree change in temperature. It is essential for calculating the dimensional changes in materials subjected to temperature variations.

Linear Expansion Formula

The linear expansion formula, expressed as ?L = ?L??T, relates the change in length (?L) to the initial length (L?), the coefficient of linear expansion (?), and the change in temperature (?T). This formula helps predict and manage the dimensional changes in objects when they are heated or cooled.

Transcript

00:01 Hello everyone, let us do the following question.

00:02 Here in this question, a cylinder is given and diameter of cylinder is given.

00:08 Diameter of cylinder is given 1 lakh centimeter and this is initially at temperature of 30 degrees celsius.

00:17 It is to be slid into a hole in a steel plate.

00:21 Here we will say that the hole has a diameter.

00:24 This is diameter of cylinder and we have to find the and it is given that.

00:30 Diameter of fold is 0 .9990 centimeter and this is also given as 30 degrees celsius to what temperature must the plate be heated we have to find the temperature and here value for alpha is given 1 .1 into 10 raised to power minus 5 degree celsius we have to tell what temperature is required we will say that the here we will solve this portion we will say that the plate it will solve for plate.

01:02 We will say the plate will expand in the same way.

01:07 The plate will expand in the same way.

01:11 The plate will expand in the same way.

01:17 Whether or not, whether or not, there is a hole in it.

01:26 There is a hole in it.

01:29 Hence the whole expands, hence the whole expands, hence the whole expands, hence the whole expands in the same way, hence the whole expands in the same way, a circle of steel, a circle of steel filling it would expand, circle of steel, feeling it would expand, feeling it would expand, feeling it would expand, feeling it would expand, we want the diameter of the whole to change by...

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