The document discusses analyzing relationships and graphs in physics. It provides examples of checking the homogeneity and validity of equations using SI units. Graphs of direct and inverse proportionalities are examined. Examples are given of interpreting the gradient and y-intercept of graphs representing physical relationships like motion, springs, and capacitors discharging. Students are asked to analyze equations, sketch graphs, and explain the physical meaning of graph properties.

1. MCA
Homogeneity of Equations
Relationships and Graphs
05.09.2021

2. MCA
Learning Objectives
11.1.1.3 - use SI base units to check the homogeneity of physical
equations;
11.1.1.5 - analyse relationships in the forms of y=axn and y=aekx;
11.1.1.6 - analyse logarithmic relationships;
11.1.2.5 - derive a relationship between two variables or recognise a
constant;

3. MCA
Homogeneity of Equations
 An equation is homogeneous only if all terms in the equation have
the same combination of SI base units
• A valid equation must be homogeneous
• A homogeneous equation is not necessarily valid
 A term of an equation is either a single number or a variable, or
numbers and variables multiplied together
5x2 – x = 3
terms

4. MCA
Homogeneity of Equations
Example: a valid and homogeneous Physics equation
 Terms in an equation can only be added or subtracted if their units are the same
Exercise
1. Is the following equation valid and homogeneous?
v = v0 + a∙t
m/s = m/s + (m/s2)∙s
m/s = m/s + m/s
𝐸𝐾 =
1
5
𝑚𝑣2
Homogeneous but not valid

5. MCA
Exercises & Problems
2. Use Newton’s equation of Universal Gravitation to determine the SI
base units of the gravitational constant G.
3. Check the validity and homogeneity of the following energy E and
mass m equation.
E = mc3
Where c is the speed of light
2. m3 kg–1 s–2; 3. Not homogeneous, therefore not valid
𝐹
𝑔 = 𝐺
𝑀𝑚
𝑟2

6. MCA
Exercises & Problems
4. Cambridge w13_qp_11
The drag coefficient Cd is a number with no units. It is used to compare
the drag on different cars at different speeds. It is given by the equation
where F is the drag force on the car, ρ is the density of the air, A is the
cross-sectional area of the car and v is the speed of the car.
What is the value of n?
A 1 B 2 C 3 D 4
B
𝐶𝑑 =
2𝐹
𝜌𝑣𝑛𝐴

7. MCA
Exercises & Problems
5. Cambridge w13_qp_13
The spring constant k of a coiled wire spring is given by the equation
where r is the radius of the wire, n is the number of turns of wire and R
is the radius of each of the turns of wire.
The quantity G depends on the material from which the wire is made.
What is a suitable unit for G?
A N m–2 B N m–1 C N m D N m2
A
𝑘 =
𝐺𝑟4
4𝑛𝑅3

8. MCA
Exercises & Problems
6. Cambridge w14_qp_21
A mass m is placed on the end of a spring that
is hanging vertically, as shown in Fig. 1.1.
The mass is made to oscillate vertically.
The time period of the oscillations of the
mass is T.
The period T is given by
𝑇 = 𝐶
𝑚
𝑘
where C is a constant and k is the spring constant.
Show that C has no units.

9. MCA
Graphs in Physics
 Graphs, and understanding the mathematical equation and physical
relationship between the dependent and independent variables are
of extreme importance in Physics and are widely used throughout
our programme
 In Physics, common graphs include direct and inverse
proportionalities, and logarithmic and exponential laws
We must understand the Mathematics and Physics behind these relationships

10. MCA
Exercises & Problems
7. a) Sketch a x-y graph which shows that
i) quantities x and y are directly proportional;
ii) quantities x and y are inversely proportional.
b) Give one example of a Physics formula for each graph on a).
8. A car is moving on a straight line with a velocity v given by
v = 4 – 2t (m s–1)
a) Sketch a graph of v as a function of time t.
b) State the gradient and the y-intercept of the graph on a).
8. b) y = 4 m/s; gradient = –2 m/s2

11. MCA
Exercises & Problems
9. The graph represents a linear relationship
between two physical quantities, x and y.
Explain the physical meaning of the gradient
in the following cases.
a) Mass; b) 1/k; c) Acceleration; d) Velocity; e) 1/R
a) Studying Newton’s Second Law, where y = force and x = acceleration.
b) Studying Hooke’s Law, where y = elongation and x = force.
c) Representing the motion of a car, where y = velocity and x = time.
d) Representing the motion of a car, where y = position and x = time.
e) Measuring the resistance, where y = current and x = p.d..
y
x

12. MCA
Exercises & Problems
10. A capacitor C discharges through a resistor R.
The voltage V across the capacitor changes with time t according to,
V0, R and C are constants and e is the natural logarithm.
a) Sketch the graph of V over t.
b) Explain how could you get a linear relationship between a function of
V and t.
c) State the meaning of the gradient and y-intercept of the graph on b).
b) lnV = lnV0 – t/RC; c) gradient = –1/RC; y = lnV0
𝑉 = 𝑉0𝑒−
𝑡
𝑅𝐶

13. MCA
Exercises & Problems
11. Explain how can you plot a straight line graph of x over t from the
equation
x = At + Bt3
A and B are constants.
x/t against t2