A pendulum clock keeps correct time at $ {0^ \circ }C$ .If the coefficient of linear expansion is $ \alpha $ , then what will be the loss in time per day , when the temperature rises by $ {t^ \circ }C$ ?
Answer
Verified
403.8k+ views
Hint: It is given in the question that the pendulum clock is accurate at $ {0^ \circ }C$ . As the temperature increases the pendulum starts expanding. The thermal coefficient of expansion is used to determine this expansion. TO find the time loss we can use the following formula:
\[\dfrac{{\Delta t}}{t} = \dfrac{1}{2}\alpha \Delta T\]
Complete answer:
In this question we are provided with the coefficient of expansion $ \alpha $ .
The rise in temperature is $ {t^ \circ }C$ .
Using the formula,
\[\dfrac{{\Delta t}}{t} = \dfrac{1}{2}\alpha \Delta T\]
\[\dfrac{{\Delta t}}{t}\] Fractional increase in time
$ \alpha $ Coefficient of linear expansion
\[\Delta T\] change in temperature = $ {t^ \circ }C$
Time in day (seconds) $ t = 24 \times 60 \times 60$
Putting all values in the formula e get,
\[
\dfrac{{\Delta t}}{t} = \dfrac{1}{2}\alpha \Delta T \\
\Rightarrow \;\dfrac{{\Delta t}}{{24 \times 60 \times 60}} = \dfrac{1}{2}\alpha (t) \\
\Rightarrow \Delta t = 43200\alpha t \\
\]
So, the time lost per day will be
\[ \Rightarrow \Delta t = 43200\alpha t\] .
Note: The loss in time per day of a pendulum may change with the metal used for making the pendulum. As the coefficient of linear expansion is dependent on the properties of metal, time lost will be different for different metal pendulums used. Additionally, the advantage of the pendulum used in clocks is that it is an harmonic oscillator which helps it swing in precise time intervals.
\[\dfrac{{\Delta t}}{t} = \dfrac{1}{2}\alpha \Delta T\]
Complete answer:
In this question we are provided with the coefficient of expansion $ \alpha $ .
The rise in temperature is $ {t^ \circ }C$ .
Using the formula,
\[\dfrac{{\Delta t}}{t} = \dfrac{1}{2}\alpha \Delta T\]
\[\dfrac{{\Delta t}}{t}\] Fractional increase in time
$ \alpha $ Coefficient of linear expansion
\[\Delta T\] change in temperature = $ {t^ \circ }C$
Time in day (seconds) $ t = 24 \times 60 \times 60$
Putting all values in the formula e get,
\[
\dfrac{{\Delta t}}{t} = \dfrac{1}{2}\alpha \Delta T \\
\Rightarrow \;\dfrac{{\Delta t}}{{24 \times 60 \times 60}} = \dfrac{1}{2}\alpha (t) \\
\Rightarrow \Delta t = 43200\alpha t \\
\]
So, the time lost per day will be
\[ \Rightarrow \Delta t = 43200\alpha t\] .
Note: The loss in time per day of a pendulum may change with the metal used for making the pendulum. As the coefficient of linear expansion is dependent on the properties of metal, time lost will be different for different metal pendulums used. Additionally, the advantage of the pendulum used in clocks is that it is an harmonic oscillator which helps it swing in precise time intervals.
Recently Updated Pages
Explain why pure liquids and solids can be ignored class 11 chemistry CBSE
At what height is the value of g half that on the surface class 11 physics CBSE
A spherical charged conductor has surface density of class 11 physics CBSE
How does onion grow and describe this process class 11 biology CBSE
The mass number of a nucleus is A Always less than class 11 chemistry CBSE
What is the difference between distillation distillation class 11 chemistry CBSE
Trending doubts
The reservoir of dam is called Govind Sagar A Jayakwadi class 11 social science CBSE
10 examples of friction in our daily life
What problem did Carter face when he reached the mummy class 11 english CBSE
Difference Between Prokaryotic Cells and Eukaryotic Cells
State and prove Bernoullis theorem class 11 physics CBSE
Proton was discovered by A Thomson B Rutherford C Chadwick class 11 chemistry CBSE