Length and Density

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Physical Quantities

A physical quantity can be measured by an instrument.
A non-physical quantity, on the other hand, refers to something we cannot develop an instrument to measure it quantitatively. (e.g. emotions)
The measurement consists of a numerical magnitude and a unit

All the physical quantities have S.I. units. S.I. units stand for International Systems of Units
S.I. units ensure that the measurements made in different parts of the world are standardised and uniform.

A) Length

Length is the distance between two points
S.I. unit for length is metre (m)
Instruments to measure length

– Measuring tape (precision of 0.1 cm)

– Metre rule (precision of 0.1 cm)

– Digital callipers/Vernier callipers (precision of 0.001 cm)

– Digital micrometre screw gauge (precision of 0.001 cm)

Metre Rule

Measures lengths from several centimetres to one metre such as the width of a book or the length of a shoe.

It has a precision of 0.1 cm

Precautions when using the metre rule

Make sure that there is no gap between the metre rule and the object to be measured.

Avoid parallax error. Parallax error in measurement occurs when the eye of the observer is not directly above the markings of the metre rule.

Avoid end error (caused by the damage of both ends of the metre rule) and zero error (when the object is not placed exactly on the zero mark of the scale)

Digital Callipers/ Vernier Callipers

Measures lengths from 1 cm up to 10cm
Precision of the vernier callipers = 0.01 m or 0.001 cm
Can be used to measure the internal and outer diameter of a bottle using the internal jaws and external jaws respectively
Tail can be used to measure depth of an object

Reading from a digital calliper*

Reading from a vernier calliper*

Step 1: From the main scale (Use the ‘0’ mark of the vernier scale as a reference point)
Step 2: From the vernier scale (Look for in line of both scales)

*Zero errors

• A zero error in vernier calliper occurs when the ‘0’ mark on the main scale is not in line with the ‘0’ mark on the vernier scale when the jaws are fully closed
• There are two types of zero error: Positive zero error and Negative zero error:

B) Volume
• Volume measures the space a matter occupies
• S.I. unit for volume: Cubic metre m³
• Instruments for measuring volume: Beaker, measuring cylinder, pipette and burette
• Only burette and pipette can measure the volume of a liquid/solution accurately
• To get an accurate reading, it must be read from the meniscus
• Meniscus is the surface of the liquid that curves at the side of the measuring instrument.

*Volume of regular shapes
• Volume of regular shapes can be found using the formulae:

Volume of irregular shapes
• The methods to measure the volume of irregular objects are as follows:
– Measuring cylinder
– Displacement cans
By using the measuring cylinder
• To measure the volume of a small heavy irregular-shaped object such as a small stone:

Procedures:
1) Record the initial volume of water
2) Submerge fully the small irregular-shaped object in water. Record the final volume.
3) The difference in these two volume readings is the volume for the small heavy irregular-shaped object.

• To measure the volume of a small light irregular-shaped object that floats in water such as a cork:

Procedures:
1) Tie the small heavy object such as a stone and submerge it fully in water. Record the initial volume of water
2) Tie the cork together with the stone and submerge fully in water. Record the final volume
3) The difference in these two volume readings is the volume for the small light irregular-shaped object.
By using the displacement can
• To measure the volume of large heavy irregular-shaped objects.

• The displacement can is filled with water to the level of the overflow vessel.
• When the irregular-shaped object is submerged in water, the water will flow out to the measuring cylinder through the overflow vessel.
• The volume of water collected in the measuring cylinder will be the volume for the large irregular-shaped object.
C) Mass
• Mass is the total amount of substance in an object
• S.I. unit for mass: Kilogram (kg)
• Instruments: Beam balance and electronic balance
• Difference between mass and weight:

D) Density
• Density is a measure of mass per unit volume of an object
• The formula to determine the density of an object:

• S.I. unit for density: Kilogram per cubic metre (kg/m³)
• Other unit: Gram per cubic centimetre (g/cm³)
• The density indicates how an bobject is being packed in a given volume. The higher the density value, the greater the compactness of the object.
Floating and sinking
• Density of an object determines if it floats or sinks in a liquid.
– An object floats in a liquid if its density is lower than the liquid.
– An object sinks in a liquid if its density is higher than the liquid.

Measuring density of an object
• The following physical quantities are needed to measure the density of an object:
– Mass of an object
– Volume of the object
• Every object has a fixed density value (E.g. Water has a density of 1g/cm³ or 1000kg/m³)
• Hence, it is possible to identify an unknown substance by its density
Length and Density (Worksheet 1)
1. Determine the unit symbols or physical quantities of each the following:

2. Calculate the following density of the following
Example: Mass of water = 100g, volume = 100 cm³
Step 1: Density of water = mass of water/volume of water
Step 2: Mass of water = 100g
Step 3: volume of water = 100cm³
Step 4: Density of water = 100/100 = 1g/cm³
(a) Mass of stone = 150g, volume of stone = 75cm³
Step 1:
Step 2:
Step 3:
Step 4:
(b) Mass of oil = 180g, Volume of oil = 200cm³
Step 1:
Step 2:
Step 3:
Step 4:
(c) Mass of glue= 210g, volume of glue = 140cm³

(d) Based on parts a to c, on the diagram below, determine which layer is oil, water and glue. Also, indicate and draw the location of the stone based on the density.

3. Calculate the density/mass/volume of the following:
(a) Mass = 25g, Density = 1.25g/cm³

(b) Mass = 225g, Density = 1.25g/cm³

(c) Volume = 70cm³, Density = 0.9g/cm³

4. Peter dropped a piece of plastic, with density of 1.50g/cm³, into a measuring cylinder that contains different liquids as shown below. Which position will the plastic be after it is dropped into the measuring cylinder?

5. A cube of side 2cm is placed in a measuring cylinder containing water as shown in the diagram below:

What is the final water level in the measuring cylinder?
Answer: __________ cm³
6. It takes 0.25 s to swing from X to Y. What is the periodof the pendulum?

7. The total time taken for the same pendulum to complete 18 oscillations is 20 s. What is the period of the pendulum?
Answer: _____s
Explain why the experiment was conducted by completing 18 oscillations and finding the average?
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8. The diagrambelow shows 4 different substances in a beaker
(a) List in order the substances shown in the diagram below from the least dense to the most dense.

(b) If a solid which is denser than Q but less dense than R is dropped into the beaker, where would it be? (Draw a square in the diagram to indicate where the solid would be).
9. A rock (rock A) with a density of 3g/cm³ is melted into molten state (liquid state). Another rock (rock B) with a density of 5g/cm³ is also melted into molten state. The two liquids are then mixed together and soldified into Rock C.
(a) If 2L of molten Rock A and 0.3L of molten rock B are mixed together, what would be the mass of Rock C?

Answer: _________g
(b) What would be the density of Rock C?

Answer: ______ g/cm³
10. The density of a substance (Solid X) is 1.7g/cm³ and the mass of solid X is 3.4g. What would be the initial reading of the measuring cylinder if the final reading after adding the solid is shown below:

11. The table below shows the densities or some substances:

(a) Name one solid that floatts in turpentine:
(b) Name one liquid that sinks in water:
(c) Name one solid that floats in Mercury but sink in water:

Length and Density (Worksheet 1B)*
1. Vernier Calliper Readings*
(a) The following vernier callipers have no zero error. Determine the reading from each gauge.

(b) Determine the type of zero error and value of the following vernier calliper measurements.

(i) Identify the type of zero error and the value.
Type = __________
value = ______________cm
When the same pair of vernier callipers is used to measure the diameter of a cup, the vernier calliper reading is as shown.

(ii) What is the diameter of the cup?
Diameter = ______ cm
2. What is the reading as shown in the figure below?

3. What is the reading for the following measurements?

4. What is the reading for the following measurements?

5. Determines the scientific notation of the following prefixes

6. Convert the following measurements into respective units.
(a) 37cm = ____ m
(b) 0.05cm = ____ m
(c) 0.05m = ____ cm
(d) 2.5kg = ____ g
(e) 2.5 Mg = ____ g
(f) 2.5 Mg = ____ kg
(g) 3.6 Km = ____ m
(h) 4.6 m = ____Gm
(i) 1250 ml = _____ l
(j) 2.6 l = ____ ml

Length and Density (Worksheet 2)
Section A: MCQ
1. Which part of the vernier callipers is used to measure the deth a beaker?
A. Internal jaws
B. External jaws
C. Tail
D. Head
2. Which of the following examples is suitable to be measured by the digital calipers?
A. Height of a person
B. Length of a chair
C. Diameter of a coin
D. Thickness of a cabinet
3. A student recorded a few measurements as follows
16.17 cm 24.123 cm 36 cm 44.3 cm
Which reading is most likely measureed by a measuring tape?
A. 16.17 cm B. 24.123 cm C. 36 cm D. 44.3 cm
4. The diagram below shows the volume of a liquid in a measuring cylinder.

What is the volume of water shown above?
A. 8.4 cm³ B. 8.5cm³ C. 8.8cm³ D. 8.9cm³

5. Which of the following best explains the parallax error when measuring an object?
A. It is an error due to the damage to the end of the measuring instrument
B. It is an error due to the gap between the measuring instrument and the object
C. It is an error due to the position of the observer’s eyes when reading the measurement off a measuring instrument.
D. It is an error due to the mishandling of the instrument by the users when measuring an object

6. What is the correct reading from a metre rule shown below if there is no end zero or zero error?

A. 5.600 cm B. 5.6 cm C. 5.60 cm D. 5 cm

7. Which of the following is the best apparatus for measuring a small irregular-shaped object?
A. Measuring cylinder B. Burette C. Pipette D. Test tube

8. The diagram below shows the length of one side of a cube.

What is the volume of the cube in m³?
A. 27 B. 2.7 x 10⁻² C. 2.7 x 10⁻⁵ D. 2.7 x 10⁻⁸

9. The diagram shows a measurement of an object using a metre rule.

The user recorded as 2.13 cm.
The user is likely to have committed
A. Positive zero error
B. Parallax error
C. End error
D. Negative zero error

Refer to the following diagram to answer questions 11 and 12.
An object with a mass of 25.5 g and density of 3.0g/cm³ is dropped into a measurong cylinders as shown in the diagram.

10. What is the volume of the object?
A. 22.5 cm³ B. 25.5 cm³ C. 8.5 cm³ D. 0.133 cm³
11. What is the final reading of volume, x cm³, shown in the measuring cylinder?
A. 8.5 cm³ B. 12.5 cm³ C. 17 cm³ D. 18.5 cm³
12. Which of the following parts of a pair of vernier callipers can be used to measure the depth of a beaker?

13. The diagram below shows the dimensions of a box.

If the box has a mass of 2.5 g, what is the density of the box?
A. 0.0625 cm³/g B. 0.1 cm³/g C. 0.625 g/cm³ D. 1.6 cm³/g

14. The mass of a plastic rod is 200 g. It has a volume of 40 cm³. What is its density in g/cm³?
A. 5000 B. 5 C. 8000 D. 80
15. The diagram below shows three different substances X, Y and Z.

Y has a density of 0.50 g/cm³. Which of the following statements is true?
A. X has a higher density than Y.
B. X has a density above 0.50 g/cm³.
C. Z has a density below 0.50 g/cm³.
D. The density of Z is higher than X.

16. The densities for three different liquids P, Q and R are given below.
Liquid P = 0.79 g/cm³ Liquid R = 1.00 g/cm³ Liquid W = 13.60 g/cm³
Which of the following statements is correct about these liquids?
A. Liquid Q floats in liquid R.
B. Liquid P always floats in liquid Q.
C. Liquid R always floats in liquid P.
D. Liquid P will be in between liquid Q and liquid R in a same container.

17. The diagram below shows a set-up to measure the volume of a small irregular-shaped stone.

If the density of the stone is 6.5 g/cm³, what is the mass of the stone?
A. 1.9 g B. 3.9 g C. 22.8 g D. 185.3 g

18. The diagram below shows a set-up to measure the volume of a ball.

The level of water containing the ball is 8.4 cm³. If the ball has a mass of 14.0 g and a density of 4.0 g/cm³, what is the original level of water x cm³, of water in the measuring cylinder?
A. 3.5 cm³ B. 4.9 cm³ C. 5.9 cm³ D. 6.5 cm³

19. An alloy is made of metals A and B according to the following masses and volumes.

What is the density of the alloy?
A. 1.67 cm³ B. 2.50 g/cm³ C. 1.79 g/cm³ D. 0.56 g/cm³

20. The diagram below shows four substances P to S, placed in a measuring cylinder.

Which substance has the greatest mass if the volume of each substance is the same?
A. P B. Q C. R D. S

21. The diagram shows a volume of an irregular-shaped object submerged completely in water.

If the object has a mass of 20 g, what is its density?
A. 2.5 g/cm³ B. 1.05 g/cm³ C. 0.4 g/cm³ D. 160 g/cm³

Section B: Short Answer Questions
1. Complete the following measurements.

2. A measuring cylinder cotains 50 cm³ of water. When a stone of mass 8 g is completely immersed in the water, the water level rises to 52 cm³.
(a) Calculate the density of the stone in g/cm³.

(b) Calculate the density of the stone in kg/m³.

(c) Describe briefly a method to determine the density of a cork which has a density lower than water.
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3. The diagram below shows a liquid in a container.