Chapter P2 - Practical skills at A Level
P2.1 Planning and analysis & P2.2 Planning
In producing your plan, you should draw a Defining the problem: identifying the variables
diagram showing the actual apparatus to be Independent variable: the quantity in an experiment that is
used, and pay particular attention to the: altered by the experimenter
• procedure to be followed Dependent variable: the quantity that is being measured and
• measurements to be taken which changes as the independent variable is altered by the
• control of variables experimenter
• analysis of the data Control variable: a quantity that has to be kept constant,
• safety precautions to be taken otherwise the relationship between the other variables is not
tested fairly
Methods of data collection
Once you have a method in mind, you should be able to describe:
• the method to be used to vary the independent variable
• how the independent variable is to be measured
• how the dependent variable is to be measured
• how other variables are to be controlled
• the arrangement of apparatus for the experiment and the procedures to be followed, with the aid of a
clear, labelled diagram.
Always check that in your account you have clearly said what you will:
• measure and how you will measure it
• change and how you will change it
• keep constant and how this is achieved
Now describe your planned experiment, making sure that you describe a logical sequence of steps to follow.
Additional details
• It is also helpful to give additional details. In particular, make sure you suggest anything that needs to be done to
ensure there is a large change in the dependent variable.
• You might also think of any difficulties in carrying out the experiment.
Safety
In some situations, the risks may be unimportant, and it may be sufficient to mention simple ideas such as wearing
goggles to protect the eyes when heating liquids or when handling stretched wires, using a safety screen, ensuring
that the apparatus is stable and not easily knocked over, using a sand tray under heavy weights to make sure that
weights do not fall on your foot and switching off currents when not in use so that wires do not overheat.
P2.3 Analysis of the data
• You need to describe how the data is used in order to reach a conclusion, and give details of any derived
quantities that are calculated.
• First, look carefully at the quantities in the relationship you have suggested (or at the formula that may be
suggested when you are given an experiment to carry out).
• If possible, you should suggest plotting a graph that you know is a straight line if the equation is correct.
You must clearly state:
• what is plotted on each axis of your graph
• that the relationship is valid if the graph gives a straight line through the origin.
You may prefer to draw a sketch graph to show what you mean, but always state clearly what type of graph you are
going to use.
More complicated analysis of data
• Need to be able to deal with quantities (take logarithms) related by equations of the form y = aˣⁿ and y = aeᵏˣ.
• Two common types of logarithms: natural logarithm (ln), logarithm too base 10 (lg).
• The ln type is more useful when dealing with an exponential formula such as eᵏˣ.
• The unit of a natural logarithm of a quantity s measured in metres is written as ln (s / m) and not as ln (s) / m or
ln (s) / ln (m). You can see that the unit is written inside the bracket with the quantity.
• E.g. y = aˣⁿ: taking logarithms on both sides: lg y = lg a + n lg x or ln y = ln a + n ln x
• E.g. y = aeᵏˣ: taking logarithms on both sides: lg y = lg a + kx or ln y = ln a + kx
, Chapter P2 - Practical skills at A Level
P2.3 Analysis of the data
Which graph to plot?
P2.4 Treatment of uncertainties
Uncertainty: an estimate of the spread of values around a measured quantity within which the true value will be found
Uncertainties and graphs
Worst acceptable line: either the steepest possible line or the shallowest possible line that passes through the error
bars of all the points
• We can use error bars to show uncertainties on graphs.
• When plotting the graph, the points are plotted as usual, and then they are extended to show the maximum and
minimum likely values, then the best fit line is drawn.
• To estimate the uncertainty in the gradient, we draw not only the best fit line but also a worst acceptable line,
passing through the extremes in the error bars.
• The gradients for both best fit and worst fit lines are calculated and the uncertainty is the difference in their
gradients: uncertainty = (gradient of best fit line) − (gradient of worst acceptable line)
Uncertainties and logarithms
• When a log graph is used and we need to include error bars, we must find the logarithm of the measured value
and the logarithm of either the largest or the smallest likely value.
• The uncertainty in the logarithm will be the difference between the two.
P2.45 Conclusion and evaluation of results
If a hypothesis is made that x is proportional to y and a straight line can be drawn from the origin through all the
error bars, then there is enough evidence here for the conclusion to be supported. If this is not possible then the
hypothesis is not validated.
, Chapter 16 - Circular motion
16.1 Describing circular motion & 16.2 Angles in radians
Key words:
Radian: the angle subtended at the centre of a circle by an arc of length equal to the radius of one circle
Angular displacement: the angle of arc through which the object has moved from its starting position
Subtended: the angle between two lines that meet at a point is said to be subtended by the two lines
To convert from degrees to radians, multiply by:
To convert from radians to degrees, multiply by:
16.3 Steady speed, changing velocity
Key words:
Speed: the rate of change of the distance moved by an object. It is a scalar quantity
Velocity: an object’s speed in a particular direction, or the rate of change of an object’s displacement. It is a vector
quantity
The arrows are straight and show the direction of motion at a particular instant. They are
drawn as tangents to the circular path. As the object travels through points A, B, C, etc., its
speed remains constant but its direction changes. Since the direction of the velocity v is
changing, it follows that v itself (a vector quantity) is changing as the object moves in a circle.
Uniform circular motion: constant magnitude of velocity, acceleration always perpendicular to velocity
16.4 Angular speed
Angular speed: the angular displacement per unit, unit: rad s ⁻¹
16.5 Centripetal forces
Key words:
Newtons’s first law (of motion): an object remains at rest or travels at constant velocity unless it is acted on by a
resultant force
Centripetal force: the resultant force on an object towards the centre of the circle when the object is rotating round
that circle at constant speed
It is important to note that the word centripetal is an
adjective. We use it to describe a force that is making
something travel along a circular path. It does not tell
us what causes this force, which might be gravitational,
electrostatic, magnetic, frictional or whatever.
The change in the velocity of the object can be determined using a vector triangle.
The vector triangle in b shows the difference between the final velocity vB and
initial velocity vA. The change in the velocity of the object between the points B and
A is shown by the smaller arrow labelled Δv. Note that the change in the velocity of
the object is (more or less):
• at right angles to the velocity at A
• directed towards the centre of the circle
The object is accelerating because its velocity changes. Since acceleration is the
rate of change of velocity, it follows that the acceleration of the object must be in
the same direction as the change in the velocity – towards the centre of the circle.
, Chapter 16 - Circular motion
16.5 Centripetal forces
Acceleration at steady speed
If the force is to make the object change its speed, it must have a component in the direction of the object’s velocity;
it must provide a push in the direction in which the object is already travelling. However, here we have a force at 90°
to the velocity, so it has no component in the required direction. (Its component in the direction of the velocity is F
cos 90° = 0.) It acts to pull the object around the circle, without ever making it speed up or slow down.
You can also use the idea of work done to show that the speed of the object moving in a circle remains the same.
The work done by a force is equal to the product of the force and the distance moved by the object in the direction of
the force. The distance moved by the object in the direction of the centripetal force is zero; hence the work done is
zero. If no work is done on the object, its kinetic energy must remain the same and hence its speed is unchanged.
Understanding circular motion
If the object is fired too slowly, gravity will pull it down towards the ground and it will
land at some distance from the cannon. A faster initial speed results in the object
landing further from the cannon.
Now, if we try a bit faster than this, the object will travel all the way round the Earth. As
the object is pulled down towards the Earth, the curved surface of the Earth falls away
beneath it. The object follows a circular path, constantly falling under gravity but never
getting any closer to the surface.
If the object is fired too fast, it travels off into space, and fails to get into a circular orbit.
So we can see that there is just one correct speed to achieve a circular orbit under
gravity. (Note that we have ignored the effects of air resistance in this discussion.)
16.6 Calculating acceleration and force
Centripetal acceleration: the acceleration of an object towards the centre of the circle when the object is rotating at
constant angular speed round that circle
The greater the mass m of the bung and the greater its speed v, the greater is the force F that is required
to spin the bung around in a circle. However, if the radius r of the circle is increased, F is smaller.
In time Δt it moves through an angle Δθ from A to B. Its speed remains
constant but its velocity changes by Δv, as shown in the vector diagram.
Since the narrow angle in this triangle is also Δθ, we can say that:
Dividing both sides of this equation by Δt and rearranging gives:
The quantity on the left is the particle’s acceleration.
The quantity on the right is the angular velocity. Substituting for these
gives:
Using v = ωr, we can eliminate ω from this equation:
Newton’s second law of motion Calculating orbital speed
Newton’s second law (of motion): the resultant We can use the force equation to calculate the speed that an
force on a body is proportional (or equal) to the object must have to orbit the Earth under gravity, as in
rate of change of momentum of the body Newton’s thought experiment. The necessary centripetal
force is provided by the Earth’s gravitational pull mg.
A force of
The radius of its orbit is equal to the
constant
Earth’s radius, approximately 6400 km.
magnitude that
is always
Both F and a are in the perpendicular to
same direction, towards the direction of
the centre of the circle. motion causes Thus, if you were to throw or
centripetal hit a ball horizontally at
acceleration. almost 8 kms⁻¹, it would go
into orbit around the Earth.
P2.1 Planning and analysis & P2.2 Planning
In producing your plan, you should draw a Defining the problem: identifying the variables
diagram showing the actual apparatus to be Independent variable: the quantity in an experiment that is
used, and pay particular attention to the: altered by the experimenter
• procedure to be followed Dependent variable: the quantity that is being measured and
• measurements to be taken which changes as the independent variable is altered by the
• control of variables experimenter
• analysis of the data Control variable: a quantity that has to be kept constant,
• safety precautions to be taken otherwise the relationship between the other variables is not
tested fairly
Methods of data collection
Once you have a method in mind, you should be able to describe:
• the method to be used to vary the independent variable
• how the independent variable is to be measured
• how the dependent variable is to be measured
• how other variables are to be controlled
• the arrangement of apparatus for the experiment and the procedures to be followed, with the aid of a
clear, labelled diagram.
Always check that in your account you have clearly said what you will:
• measure and how you will measure it
• change and how you will change it
• keep constant and how this is achieved
Now describe your planned experiment, making sure that you describe a logical sequence of steps to follow.
Additional details
• It is also helpful to give additional details. In particular, make sure you suggest anything that needs to be done to
ensure there is a large change in the dependent variable.
• You might also think of any difficulties in carrying out the experiment.
Safety
In some situations, the risks may be unimportant, and it may be sufficient to mention simple ideas such as wearing
goggles to protect the eyes when heating liquids or when handling stretched wires, using a safety screen, ensuring
that the apparatus is stable and not easily knocked over, using a sand tray under heavy weights to make sure that
weights do not fall on your foot and switching off currents when not in use so that wires do not overheat.
P2.3 Analysis of the data
• You need to describe how the data is used in order to reach a conclusion, and give details of any derived
quantities that are calculated.
• First, look carefully at the quantities in the relationship you have suggested (or at the formula that may be
suggested when you are given an experiment to carry out).
• If possible, you should suggest plotting a graph that you know is a straight line if the equation is correct.
You must clearly state:
• what is plotted on each axis of your graph
• that the relationship is valid if the graph gives a straight line through the origin.
You may prefer to draw a sketch graph to show what you mean, but always state clearly what type of graph you are
going to use.
More complicated analysis of data
• Need to be able to deal with quantities (take logarithms) related by equations of the form y = aˣⁿ and y = aeᵏˣ.
• Two common types of logarithms: natural logarithm (ln), logarithm too base 10 (lg).
• The ln type is more useful when dealing with an exponential formula such as eᵏˣ.
• The unit of a natural logarithm of a quantity s measured in metres is written as ln (s / m) and not as ln (s) / m or
ln (s) / ln (m). You can see that the unit is written inside the bracket with the quantity.
• E.g. y = aˣⁿ: taking logarithms on both sides: lg y = lg a + n lg x or ln y = ln a + n ln x
• E.g. y = aeᵏˣ: taking logarithms on both sides: lg y = lg a + kx or ln y = ln a + kx
, Chapter P2 - Practical skills at A Level
P2.3 Analysis of the data
Which graph to plot?
P2.4 Treatment of uncertainties
Uncertainty: an estimate of the spread of values around a measured quantity within which the true value will be found
Uncertainties and graphs
Worst acceptable line: either the steepest possible line or the shallowest possible line that passes through the error
bars of all the points
• We can use error bars to show uncertainties on graphs.
• When plotting the graph, the points are plotted as usual, and then they are extended to show the maximum and
minimum likely values, then the best fit line is drawn.
• To estimate the uncertainty in the gradient, we draw not only the best fit line but also a worst acceptable line,
passing through the extremes in the error bars.
• The gradients for both best fit and worst fit lines are calculated and the uncertainty is the difference in their
gradients: uncertainty = (gradient of best fit line) − (gradient of worst acceptable line)
Uncertainties and logarithms
• When a log graph is used and we need to include error bars, we must find the logarithm of the measured value
and the logarithm of either the largest or the smallest likely value.
• The uncertainty in the logarithm will be the difference between the two.
P2.45 Conclusion and evaluation of results
If a hypothesis is made that x is proportional to y and a straight line can be drawn from the origin through all the
error bars, then there is enough evidence here for the conclusion to be supported. If this is not possible then the
hypothesis is not validated.
, Chapter 16 - Circular motion
16.1 Describing circular motion & 16.2 Angles in radians
Key words:
Radian: the angle subtended at the centre of a circle by an arc of length equal to the radius of one circle
Angular displacement: the angle of arc through which the object has moved from its starting position
Subtended: the angle between two lines that meet at a point is said to be subtended by the two lines
To convert from degrees to radians, multiply by:
To convert from radians to degrees, multiply by:
16.3 Steady speed, changing velocity
Key words:
Speed: the rate of change of the distance moved by an object. It is a scalar quantity
Velocity: an object’s speed in a particular direction, or the rate of change of an object’s displacement. It is a vector
quantity
The arrows are straight and show the direction of motion at a particular instant. They are
drawn as tangents to the circular path. As the object travels through points A, B, C, etc., its
speed remains constant but its direction changes. Since the direction of the velocity v is
changing, it follows that v itself (a vector quantity) is changing as the object moves in a circle.
Uniform circular motion: constant magnitude of velocity, acceleration always perpendicular to velocity
16.4 Angular speed
Angular speed: the angular displacement per unit, unit: rad s ⁻¹
16.5 Centripetal forces
Key words:
Newtons’s first law (of motion): an object remains at rest or travels at constant velocity unless it is acted on by a
resultant force
Centripetal force: the resultant force on an object towards the centre of the circle when the object is rotating round
that circle at constant speed
It is important to note that the word centripetal is an
adjective. We use it to describe a force that is making
something travel along a circular path. It does not tell
us what causes this force, which might be gravitational,
electrostatic, magnetic, frictional or whatever.
The change in the velocity of the object can be determined using a vector triangle.
The vector triangle in b shows the difference between the final velocity vB and
initial velocity vA. The change in the velocity of the object between the points B and
A is shown by the smaller arrow labelled Δv. Note that the change in the velocity of
the object is (more or less):
• at right angles to the velocity at A
• directed towards the centre of the circle
The object is accelerating because its velocity changes. Since acceleration is the
rate of change of velocity, it follows that the acceleration of the object must be in
the same direction as the change in the velocity – towards the centre of the circle.
, Chapter 16 - Circular motion
16.5 Centripetal forces
Acceleration at steady speed
If the force is to make the object change its speed, it must have a component in the direction of the object’s velocity;
it must provide a push in the direction in which the object is already travelling. However, here we have a force at 90°
to the velocity, so it has no component in the required direction. (Its component in the direction of the velocity is F
cos 90° = 0.) It acts to pull the object around the circle, without ever making it speed up or slow down.
You can also use the idea of work done to show that the speed of the object moving in a circle remains the same.
The work done by a force is equal to the product of the force and the distance moved by the object in the direction of
the force. The distance moved by the object in the direction of the centripetal force is zero; hence the work done is
zero. If no work is done on the object, its kinetic energy must remain the same and hence its speed is unchanged.
Understanding circular motion
If the object is fired too slowly, gravity will pull it down towards the ground and it will
land at some distance from the cannon. A faster initial speed results in the object
landing further from the cannon.
Now, if we try a bit faster than this, the object will travel all the way round the Earth. As
the object is pulled down towards the Earth, the curved surface of the Earth falls away
beneath it. The object follows a circular path, constantly falling under gravity but never
getting any closer to the surface.
If the object is fired too fast, it travels off into space, and fails to get into a circular orbit.
So we can see that there is just one correct speed to achieve a circular orbit under
gravity. (Note that we have ignored the effects of air resistance in this discussion.)
16.6 Calculating acceleration and force
Centripetal acceleration: the acceleration of an object towards the centre of the circle when the object is rotating at
constant angular speed round that circle
The greater the mass m of the bung and the greater its speed v, the greater is the force F that is required
to spin the bung around in a circle. However, if the radius r of the circle is increased, F is smaller.
In time Δt it moves through an angle Δθ from A to B. Its speed remains
constant but its velocity changes by Δv, as shown in the vector diagram.
Since the narrow angle in this triangle is also Δθ, we can say that:
Dividing both sides of this equation by Δt and rearranging gives:
The quantity on the left is the particle’s acceleration.
The quantity on the right is the angular velocity. Substituting for these
gives:
Using v = ωr, we can eliminate ω from this equation:
Newton’s second law of motion Calculating orbital speed
Newton’s second law (of motion): the resultant We can use the force equation to calculate the speed that an
force on a body is proportional (or equal) to the object must have to orbit the Earth under gravity, as in
rate of change of momentum of the body Newton’s thought experiment. The necessary centripetal
force is provided by the Earth’s gravitational pull mg.
A force of
The radius of its orbit is equal to the
constant
Earth’s radius, approximately 6400 km.
magnitude that
is always
Both F and a are in the perpendicular to
same direction, towards the direction of
the centre of the circle. motion causes Thus, if you were to throw or
centripetal hit a ball horizontally at
acceleration. almost 8 kms⁻¹, it would go
into orbit around the Earth.
