You have been provided with a metre rule, a clamp, and a set of masses. Clamp the metre rule to the edge of the bench such that 90cm of the rule projects fr...
Question 1 Report
You have been provided with a metre rule, a clamp, and a set of masses.
- Clamp the metre rule to the edge of the bench such that 90cm of the rule projects from the edge as shown in the diagram above. Ensure that the rule is capable of performing oscillatory motion.
- Fix a mass M = 50g at the free end of the rule.
- Deflect the rule slightly such that it performs vertical oscillation.
- Determine the time t for 10 complete oscillations.
- Calculate the period T of the oscillations and evaluate T\(^{2}\)
- Repeat the procedure for four other values of M = 100, 150, 200, and 250g. In each case determine and record the corresponding values of t, T, and T\(^{2}\). Tabulate your readings.
- Plot a graph of T\(^{2}\) on the vertical axis against M on the horizontal axis, starting both axes from the origin (0,0).
- Determine the slopes, of the graph and its intercept C on the vertical axis.
- Evaluate k = 4\(\pi\)/s. [Take \(\pi\) = \(\frac{22}{7}\)].
- From your graph, determine the period T, when M= 180g.
- State two precautions taken to ensure accurate results.
(b)i. Explain simple harmonic motion.
ii. Define period and frequency, with respect to a simple harmonic motion.
Precautions:
- l ensured that the metre rule was firmly clamped
- Readings were repeated
- Parallax was avoided when readings on the stopwatch/clock were taken.
- zero error was noted and corrected on the stopwatch/clock.
(b)i. Simple harmonic motion is a motion in which the acceleration is proportional to the displacement from a fixed point and is directed towards the point.
ii. Period is the time taken by an oscillatory body to make one complete oscillation.
Frequency: is the number of complete oscillations performed in one second.
Answer Details
This is a loaded-cantilever oscillation experiment. The metre rule projecting 90 cm from the bench behaves like a spring; when a mass \(M\) fixed at its free end is deflected and released, it performs vertical simple harmonic motion. For each mass you time 10 complete oscillations, find the period \(T=\dfrac{t}{10}\), and compute \(T^2\). Theory gives \(T^2=\dfrac{4\pi^2}{k}M + C\), so a graph of \(T^2\) against \(M\) is a straight line whose slope allows \(k\) to be found.
Readings and table of values
| M (g) | t (s) for 10 oscillations | T = t/10 (s) | T2 (s2) |
|---|---|---|---|
| 50 | 3.16 | 0.316 | 0.100 |
| 100 | 3.87 | 0.387 | 0.150 |
| 150 | 4.47 | 0.447 | 0.200 |
| 200 | 5.00 | 0.500 | 0.250 |
| 250 | 5.48 | 0.548 | 0.300 |
Graph of T2 against M
Slope and intercept
Using two convenient points on the line, \((M_1,T_1^2)=(50,\,0.100)\) and \((M_2,T_2^2)=(250,\,0.300)\):
\[ s=\frac{\Delta T^2}{\Delta M}=\frac{0.300-0.100}{250-50}=\frac{0.200}{200}=1.0\times10^{-3}\ \text{s}^2\,\text{g}^{-1} \]The line meets the vertical axis where \(M=0\), giving the intercept
\[ C = 0.05\ \text{s}^2 \]Evaluation of k
From \(T^2=\dfrac{4\pi^2}{k}M + C\), the slope is \(s=\dfrac{4\pi^2}{k}\), so \(k=\dfrac{4\pi^2}{s}\). With \(\pi=\dfrac{22}{7}\):
\[ k=\frac{4\pi^2}{s}=\frac{4\times\left(\frac{22}{7}\right)^2}{1.0\times10^{-3}}=\frac{4\times 9.878}{1.0\times10^{-3}}=\frac{39.51}{1.0\times10^{-3}}\approx 3.95\times10^{4} \]Period when M = 180 g
Reading up from \(M=180\ \text{g}\) to the line and across to the vertical axis gives \(T^2=0.23\ \text{s}^2\), so
\[ T=\sqrt{0.23}=0.48\ \text{s} \]Two precautions
- The metre rule was firmly clamped to the bench so that only the projecting part oscillated.
- Parallax error was avoided when reading the stopwatch, the zero error was noted and corrected, and each timing was repeated and averaged.
(b)(i) Simple harmonic motion
Simple harmonic motion is the motion of a body in which its acceleration is directly proportional to its displacement from a fixed point and is always directed towards that fixed point.
(b)(ii) Period and frequency
The period is the time taken by the oscillating body to make one complete oscillation. The frequency is the number of complete oscillations made in one second, related to the period by \(f=\dfrac{1}{T}\).
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