21 The time period , T seconds, of a simple pendulum of length 7cm is given by the formula T=2pi sqrt ((l)/(g)) Katie uses a simple pendulum in an experiment to find an estimate for the value of g. Here are her results. l=52.0 correct to 3 significant figures. T=1.45 correct to 3 significant figures. Work out the upper bound and the lower bound for the value of g. Use pi =3.142 You must show all your working.

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21 The time period , T seconds, of a simple pendulum of length 7cm is given by the formula T=2pi sqrt ((l)/(g)) Katie uses a simple pendulum in an experiment to find an estimate for the value of g. Here are her results. l=52.0 correct to 3 significant figures. T=1.45 correct to 3 significant figures. Work out the upper bound and the lower bound for the value of g. Use pi =3.142 You must show all your working.

Answer 1

The upper bound for the value of \(g\) is approximately \(98.32 \, \text{cm/s}^2\) and the lower bound for the value of \(g\) is approximately \(96.73 \, \text{cm/s}^2\).

Answer 2

## Step 1: ### The formula for the period \(T\) of a simple pendulum is given by \(T = 2\pi \sqrt{\frac{l}{g}}\), where \(l\) is the length of the pendulum and \(g\) is the acceleration due to gravity. ## Step 2: ### To find \(g\), we rearrange the equation to solve for \(g\). Squaring both sides, we get: \[T^2 = (2\pi)^2 \frac{l}{g}\] ### Simplifying, we obtain: \[T^2 = 4\pi^2 \frac{l}{g}\] ### Rearranging to solve for \(g\), we get: \[g = \frac{4\pi^2 l}{T^2}\] ## Step 3: ### Given \(l = 52.0\) cm and \(T = 1.45\) seconds, we first find the upper and lower bounds for \(l\) and \(T\). ### For \(l\): #### The upper bound is \(52.0 + 0.05 = 52.05\) cm. #### The lower bound is \(52.0 - 0.05 = 51.95\) cm. ### For \(T\): #### The upper bound is \(1.45 + 0.005 = 1.455\) seconds. #### The lower bound is \(1.45 - 0.005 = 1.445\) seconds. ## Step 4: ### Calculate the upper bound for \(g\) using the lower bound of \(T\) and the upper bound of \(l\): \[g_{\text{upper}} = \frac{4 \cdot (3.142)^2 \cdot 52.05}{(1.445)^2}\] ## Step 5: ### Calculate the lower bound for \(g\) using the upper bound of \(T\) and the lower bound of \(l\): \[g_{\text{lower}} = \frac{4 \cdot (3.142)^2 \cdot 51.95}{(1.455)^2}\] ## Step 6: ### Perform the calculations: \[g_{\text{upper}} = \frac{4 \cdot 9.869 \cdot 52.05}{2.087025} \approx 98.32\] \[g_{\text{lower}} = \frac{4 \cdot 9.869 \cdot 51.95}{2.115025} \approx 96.73\]

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