Question

At a certain temperature, a simple pendulum has a period of 1.500 seconds. The support wire is made of brass and has a coefficient of linear thermal expansion of 1.90 × 10–5/C°. How much must the temperature be increased to increase the period to 1.506 seconds?

Answer 1

Answer:

The temperature must be increased by 471.19° to increases the time period from 1.500 s to 1.506 s.

Explanation:

Given that,

Time period = 1.500 sec

Linear thermal expansion $\alpha= 1.90\times10^{-5}\ /C^{\circ}$

Increases period = 1.506 sec

We need to calculate the initial length

Using formula of time period

$T=2\pi\sqrt{\dfrac{l}{g}}$

Put the value into the formula

$1.500=2\pi\sqrt{\dfrac{l}{9.8}}$

On squaring both side

$(1.500)^2=4\pi^2\times\dfrac{l}{9.8}$

$l=\dfrac{1.500^2\times9.8}{4\pi^2}$

$l=0.5585\ m$

We need to calculate the new length

For period to be 1.506

$1.506=2\pi\sqrt{\dfrac{l'}{9.8}}$

$l'=\dfrac{1.506^2\times9.8}{4\pi^2}$

$l'=0.5635\ m$

We need to calculate the temperature

Using formula of thermal expansion

$l'=l(1+\alpha T)$

$T=\dfrac{l'-l}{l\alpha}$

Put the value into the formula

$T=\dfrac{0.5635-0.5585}{0.5585\times 1.90\times10^{-5}}$

$T=471.19^{\circ}$

Hence, The temperature must be increased by 471.19° to increases the time period from 1.500 s to 1.506 s.