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Question

The time period of a simple pendulum is given by T=2πlg The measured value of the length of the pendulum is 10cm known to a 1mm accuracy. The time for 200 oscillations of the pendulum is found to be 100 seconds using a clock of 1s resolution. The percentage accuracy in the determination of g using this pendulum is x. The value of x to the nearest integer is.


A

5%

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B

4%

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C

3%

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D

2%

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Solution

The correct option is C

3%


Step 1: Given Data:

The measured value of the length of the pendulum l=10cm=10×10-2m

Accuracy in lengthl=1mm=10-3m

The time for 200 oscillations of the pendulum t=100 seconds

Time period T=TimetakenbypendulumNoofoscillation=100200=0.5

Resolution of clockt=1s

Accuracy in the time period T=1200=5×10-3

The percentage accuracy in the determination of gis x

Step 2: Formula Used:

The given formula

T=2πlg

Where l=length, T=time, and g=gravity

Step 3: Calculating “g”

The given formula

T=2πlg

On squaring both the sides

T2=4π2lg

Rearranging the formula to calculate the value of g

g=4π2lT21

Step 3: Calculating the percentage accuracy

The maximum accuracy in the determination of g, from equation (1)

g=4π2lT2

Δgg=Δll+2ΔTTΔgg=10310×102+2×5×10-30.5Δgg=0.01+0.02Δgg=0.03

The percentage accuracy

gg×100=0.03×100x=3%

Hence option C is the correct answer.


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