| Date | May 2015 | Marks available | 2 | Reference code | 15M.2.SL.TZ2.1 |
| Level | Standard level | Paper | Paper 2 | Time zone | Time zone 2 |
| Command term | Calculate, Deduce, Determine, and Draw | Question number | 1 | Adapted from | N/A |
Question
Data analysis question.
A simple pendulum of length l consists of a small mass attached to the end of a light string.
The time T taken for the mass to swing through one cycle is given by
\[T = 2\pi \sqrt {\frac{l}{g}} \]
where g is the acceleration due to gravity.
A student measures T for one length l to determine the value of g. Time \(T = 1.9s \pm 0.1s\) and length \(l = 0.880m \pm 0.001m\). Calculate the fractional uncertainty in g.
The student modifies the simple pendulum of length L so that, after release, it swings for a quarter of a cycle before the string strikes a horizontal thin edge. For the next half cycle, the pendulum swings with a shorter length x. The string then leaves the horizontal thin edge to swing with its original length L.
The length L of the string is kept constant during the experiment. The vertical position of the horizontal thin edge is varied to change x.
The graph shows the variation of the time period with \(\sqrt x \) for data obtained by the student together with error bars for the data points. The error in \(\sqrt x \) is too small to be shown.
(i) Deduce that the time period for one complete oscillation of the pendulum is given by
\[T = \frac{\pi }{{\sqrt g }}\left( {\sqrt L + \sqrt x } \right)\]
(ii) On the graph, draw the best-fit line for the data.
(iii) Determine the gradient of the graph.
(iv) State the value of the intercept on the T-axis.
(v) The equation of a straight line is \(y = mx + c\). Determine, using your answers to (b)(iii) and (b)(iv), the intercept on the \(\sqrt {\rm{x}} \)-axis.
(vi) Calculate L.
Markscheme
(i) half of cycle takes \(\pi \sqrt {\frac{L}{g}} \) other half takes \(\pi \sqrt {\frac{x}{g}} \) and combine to give result;
\[\left( {\frac{\pi }{g}\left( {\sqrt L + \sqrt x } \right)} \right)\]
(ii) 
straight line of any length through all error bars;
Do not accept kinked, fuzzy, doubled lines.
(iii) more than half line of their line used for gradient determination;
read-offs correct;
correct working leading to their gradient; (best straight line gives 1.03)
At least two significant figures are required in answer.
(iv) their intercept ± half a square; (best straight line gives 0.96 s)
(v) makes correct substitution for T=0;
correct answer from own data including negative sign;} (unit not required -0.93m\(\frac{1}{2}\))
Allow substitution into equation for straight line, but data point used must lie on candidate line.
N.B. x in \(y = mx + c\) is \(\sqrt {\rm{x}} \) on the axis – give BOD if not clear but answer correct.
