INNOVA JUNIOR COLLEGE
JC1 Physics (2009)
Tutorial 1: Measurement Worksheet
Uncertainty of a Derived Quantity
For a reading (a ± ∆a):
a is the experimental reading
∆a is the uncertainty in the reading
For multiplication and division operations, add fractional uncertainties.
y b c
i.e. y = b × c


or y = b ÷ c,
y
b
c
Therefore
bm
If y  n
c
b  b  ... m times 

c  c  ... n times 
y
b
c
m
n
y
b
c
Example 10 (N98/I/2 modified)
The density of a rectangular block was determined by measuring the mass and linear
dimensions of the block.
mass
= (25.0  0.1) g
length = (5.00  0.01) cm
breadth = (2.00  0.01) cm
height = (1.00  0.01) cm
How should density of the material be recorded as? The yellow-highlighted are the answer to
the examples.
1. Value of variable: Make the physical quantity the subject of the equation.
m m

V lbh
25.0

5.00  2.00  1.00
 2.50 g cm 3

Do not round off the answer yet. It is alright to leave your answer 6/7 s.f. if necessary.
2. Uncertainty equation: Since the operation is multiplication or division operation of
variable, add fractional uncertainties.



m
lbh

m l b h



m
l
b
h
1
3. Uncertainty of variable: substitute values into the uncertainty equation
m = (25.0  0.1) mm


0.1 0.01 0.01 0.01



25.0 5.00 2.00 1.00


 0.021
2.50
  0.0525
 0.05 g cm-3 (1 s.f.)
Since the value is a uncertainty, the value of the uncertainty cannot be too precise. Thus the
value of the uncertainty is ALWAYS rounded up to 1 s.f.
4. Final answer: Same d.p. for the calculated value and its associated uncertainty.
   2.50  0.05 g cm-3
Always write the absolute uncertainty 1st, then write the value of the derived quantity. The
value of the derived quantity follows the d.p. of the uncertainty.
In this case, ρ is rounded off to 2 d.p. as the precision of Δ ρ is 2 d.p.
Section 1: Identifying the correct uncertainty expression
L
g
The associated uncertainty of g can be found using:
(a) g  2L  T
(b) g  2L  T
g T L


(c)
g
T
L
g 1 T 1 L


(d)
g
2 T
2 L
1. T  2
2. x 
p
q3
The associated uncertainty of x can be found using:
(a)
x

x
p  q 


p  q 
3
(b) x  p   q 
3
 q 
x 1 p

 3

x
2 p
 q 
1
(d) x  p  3q
2
(c)
2
2
m r2
5
The associated uncertainty of I can be found using:
2
2
(a) I  m   r 
5
2
I 2 m  r 

 
(b)
I
5 m  r 
I m
r
(c)

2
I
m
r
(d) I  m  2r
3. I 
Section 2: Guided Example
4. The radius of a solid sphere is measured to be (6.5 ± 0.2) cm. Determine the volume of the
sphere with its uncertainty.
Write the equation for V:
V
4 3
r
3
Determine the value of V:
V
Write the uncertainty equation for ΔV:
4 3
r
3
V


V
V
Hence, substitute values into the uncertainty equation to determine ΔV:
Final answer:
V = (V ± ∆V)
=(
±
) cm3
The precision of V is recorded to the d.p. of ∆V.
Ans: (1200 ± 100) cm3
3
5. A resistor is marked as having a value of 5.9 Ω ± 2%. The power P dissipated in the resistor,
when connected in a simple electrical circuit, was to be calculated from the current in the
resistor, which measured as (1.40 ± 0.05) mA. What is the value of calculated P together
with its associated uncertainty?
Write the equation for P:
P = I2R
Determine the value of P:
P=
Write the uncertainty equation for ΔP:
P = I2R
P

P
Hence substitute values into the uncertainty equation to determine ΔP:
Final answer:
P = (P ± ∆P)
=(
±
) × 10-6 cm3
The precision of P is recorded to the d.p. of ∆P.
Ans: (12 ± 1) × 10-6 cm3
6. A car accelerates uniformly from rest and travels a distance of (100 ± 1) m. If the
acceleration of the car is (6.5 ± 0.5) m s-2, what would be its final velocity, together with its
associated uncertainty, at the end of the distance covered?
Ans: (36 + 2) m s-1
4