Exercises: Analysing experimental data

A student in a school in Kildare verifies Hooke's law by hanging masses from a spring and measuring its extension.

Mass $m$ / kg	Extension $x$ / m
0.10	0.020
0.20	0.041
0.30	0.059
0.40	0.081

1. a) Calculate the force on the spring for each mass, using $g = 9.8\ \text{m s}^{-2}$.
2. b) Calculate the ratio $F/x$ for each reading.
3. c) State what the results suggest about Hooke's law.

A student measures the period $T$ of a simple pendulum of length $l = 1.00\ \text{m}$ by timing $20$ complete oscillations, which take $40.4\ \text{s}$.

1. a) Calculate the period of the pendulum.
2. b) Using $T=2\pi\sqrt{l/g}$, calculate the value of $g$.
3. c) Explain why timing $20$ oscillations is better than timing one.

A student determines the density of a metal block. Its mass is $0.2160\ \text{kg}$, and its dimensions are $6.0\ \text{cm}\times 3.0\ \text{cm}\times 4.0\ \text{cm}$.

1. a) Calculate the volume of the block in $\text{m}^3$.
2. b) Calculate the density in $\text{kg m}^{-3}$.
3. c) Suggest, from the table of densities of metals in the data booklet, a metal that this block could be made from, and state the property you compared.

A student verifies $F_{net}=ma$ using a trolley of total mass $0.80\ \text{kg}$ on a runway with friction compensated. The trolley is pulled by different forces and the acceleration is found using light gates.

Force $F$ / N	Acceleration $a$ / $\text{m s}^{-2}$
0.20	0.26
0.40	0.49
0.60	0.77
0.80	1.02
1.00	1.24

1. a) Describe the graph that should be plotted and the shape expected.
2. b) Calculate the gradient from the first and last data points.
3. c) Use the gradient to calculate the mass of the trolley, and calculate the percentage difference from $0.80\ \text{kg}$.
4. d) Give one reason for the difference.

Students investigate the period $T$ of a pendulum for different lengths $l$.

$l$ / m	$T$ / s
0.20	0.90
0.40	1.27
0.60	1.55
0.80	1.79
1.00	2.01

1. a) Calculate $T^2$ for each length.
2. b) Calculate the gradient of a graph of $T^2$ against $l$ using the first and last points.
3. c) Use the gradient to calculate $g$, given that $T^2=\dfrac{4\pi^2}{g}l$.
4. d) State why the amplitude of swing should be kept small.

A student investigates conservation of momentum with two trolleys on a runway. Trolley A has mass $0.50\ \text{kg}$ and moves at $0.80\ \text{m s}^{-1}$ towards trolley B, of mass $0.50\ \text{kg}$, which is stationary. They stick together on impact. The student measures the velocity after the collision as $0.37\ \text{m s}^{-1}$.

1. a) Calculate the total momentum before the collision.
2. b) Calculate the total momentum after the collision using the measured velocity.
3. c) Calculate the percentage difference and state whether momentum is conserved within experimental error.
4. d) Calculate the kinetic energy before and after, and explain the difference.

A student measures the speed of sound in air by the resonance of a tuning fork of frequency $512\ \text{Hz}$ above a tube partly filled with water. The first resonance is found when the air column is $16.0\ \text{cm}$ long, and the second resonance at $49.5\ \text{cm}$.

1. a) Calculate the wavelength using the difference between the two resonance lengths.
2. b) Calculate the speed of sound.
3. c) Use the first resonance length to find the end correction $e$, given that $L_1+e=\lambda/4$.
4. d) Explain why the use of two resonances gives a better value than the use of one, and estimate the percentage uncertainty if each length has an uncertainty of $\pm 0.2\ \text{cm}$.

A student uses a ball dropped from different heights $h$ and a light-gate timer to measure the time $t$ to fall, and then finds $g$.

$h$ / m	$t$ / s
0.20	0.203
0.40	0.286
0.60	0.350
0.80	0.404
1.00	0.452

1. a) Show that $h=\tfrac12 g t^2$ means a plot of $h$ against $t^2$ is a straight line through the origin, and give the gradient in terms of $g$.
2. b) Calculate $t^2$ for each point and the gradient from the first and last points.
3. c) Calculate $g$ and the percentage difference from $9.81\ \text{m s}^{-2}$.
4. d) Discuss the main source of error and how it would affect the intercept of the graph.

A student verifies conservation of energy for a trolley rolling down a low-friction ramp. The trolley has mass $0.60\ \text{kg}$. It starts from rest and descends a vertical height of $0.15\ \text{m}$. A light gate measures its speed at the bottom as $1.62\ \text{m s}^{-1}$.

1. a) Calculate the loss in gravitational potential energy.
2. b) Calculate the gain in kinetic energy.
3. c) Calculate the percentage of energy lost and suggest two possible sources of this loss.
4. d) The light gate measures the speed using a card of length $5.0\ \text{cm}$ and a time of $0.0309\ \text{s}$. Explain why this gives an average speed rather than the speed at the bottom, and how this affects the result.