Question

A pendulum on a grandfather clock

is supposed to oscillate once every
2.00 s, but actually oscillates once
every 1.99 s. How much must you
increase its length to correct its
period to 2.00 s?
(Unit = m)

Answer 1

Answer:.0099

Explanation:accelus

Answer 2

you have to increase 0.01 meters the length

Step-by-step explanation

The period of a pendulum is given by:

`T=2\text{ }\pi\sqrt[]{(L)/(g)}`

where T is the period, L is the length , a g is the the acceleration of the gravity (9.8 m per square second)

s

Step 1

find the length, for period= 2 s

Let

`T_1=2\text{ seconds}`

with this value, we can find the length

replace, and isolate L

`\begin{gathered} T=2\text{ }\pi\sqrt[]{(L)/(g)} \\ 2=2\text{ }\pi\sqrt[]{(L)/(g)} \\ \text{divide both sides by 2}\pi \\ (2)/(2\pi)=\frac{2\text{ }\pi}{2\pi}\sqrt[]{(L)/(g)} \\ (1)/(\pi)=\sqrt[]{(L)/(g)} \\ ((1)/(\pi))^2=(\sqrt[]{(L)/(g)})^2 \\ (1)/(\pi^2)=(L)/(g) \\ \text{Multiply both sides by g} \\ (1)/(\pi^2)\cdot g\cdot g=(L)/(g) \\ (g)/(\pi^2)=L \end{gathered}`

so, when the period is 2.00 s the length of the pendulum is

`\begin{gathered} L_1=(g)/(\pi^2) \\ L_1=(9.8)/(\pi^2) \\ L_1=0.9929475997\text{ m} \end{gathered}`

Step 2

now, for Period = 1.99 s

Let

`T=1.99\text{ s}`

replace and solve for L

`\begin{gathered} T=2\text{ }\pi\sqrt[]{(L)/(g)} \\ 1.99=2\text{ }\pi\sqrt[]{(L)/(g)} \\ \text{divide both sides by 2}\pi \\ (1.99)/(2\pi)=\frac{2\text{ }\pi}{2\pi}\sqrt[]{(L)/(g)} \\ (1.99)/(2\pi)=\sqrt[]{(L)/(g)} \\ ((1.99)/(2\pi))^2=(\sqrt[]{(L)/(g)})^2 \\ \\ 0.1003105048=(L)/(g) \\ \text{Multiply both sides by g} \\ 0.1003105048\cdot g=(L)/(g)\cdot g \\ 0.1003105048\cdot g=L \\ L=0.1003105048\cdot9.8 \\ L_2=0.983049474\text{ m} \end{gathered}`

so, when the period is 1.99 s , the length is 0.9830429474 m

Step 3

finally, to know how much you must increase the length, subtract L2 from L1

so

`\begin{gathered} L_1-L_2=0.9929475997\text{ m-}0.9830429474m \\ L_1-L_2\approx0.99-0.98 \\ L_1-L_2\approx0.01 \end{gathered}`

therefore, the make the pendulum osscilates with period of 2 seconds, you have to increase 0.01 meters the length

I hope this helps you