0625 · 1.1Physical quantities and measurement techniques

Measuring length and volume

0:00A ruler can measure a book easily, but one sheet of paper is much harder to measure. A stopwatch can time a race, but timing one quick swing introduces a different difficulty. In both cases, measuring several together can give a better result. This video explains careful measurement, then shows why some physical quantities also need a direction. A physical quantity is something measurable, such as length, time or force.

0:24A measurement normally needs both a numerical value and a unit. Saying a pencil is fifteen long leaves its length unclear. Saying fifteen centimetres gives the number a meaning, because the unit tells us the scale being used. To measure length, place a ruler alongside the object, with its scale parallel to the length being measured. Align one end with the zero mark, rather than assuming the ruler's edge is zero.

0:48Read the position of the other end. Look straight down at the scale so your viewing angle does not shift the apparent position. This viewing error is called parallax error. If the zero mark is damaged, start at another clear mark and subtract the initial reading from the final reading. Suppose a pencil starts at two centimetres and ends at seventeen centimetres. Its length is 17 − 2, which equals fifteen centimetres.

1:12For liquid volume, use a measuring cylinder standing upright on a level surface. First check the units and the value represented by each small division. Water usually has a curved upper surface called a meniscus. Read the bottom of this meniscus with your eye at the same height, avoiding parallax error. A measuring cylinder can also measure the volume of an irregular solid by water displacement. Use an object that sinks and does not dissolve or react with the water.

1:37Record the initial water volume, then lower the object until it is completely submerged. Remove trapped air bubbles and record the new reading, keeping all the water inside the cylinder. Imagine the initial reading is thirty cubic centimetres and the final reading is forty-two cubic centimetres. The object occupies space that the water previously occupied, so the water level rises. The object's volume equals the final reading minus the initial reading.

2:0342 − 30 gives a volume of twelve cubic centimetres. The final reading alone is not the object's volume, because it includes the original water. Measuring cylinders may be marked in millilitres, and one millilitre equals one cubic centimetre. Always check the scale rather than assuming each division represents one unit. Choose a suitably sized cylinder with divisions you can read clearly.

Measuring multiples accurately

2:26Now return to the paper problem. One sheet is too thin for an ordinary ruler to measure reliably, but a stack produces a readable thickness. Stack one hundred sheets neatly, without gaps or loose folds, and measure their total thickness. Dividing this total by the number of sheets gives the average thickness of one sheet. Suppose one hundred sheets have a total thickness of ten millimetres. Average thickness equals total thickness divided by the number of sheets.

2:5210 ÷ 100 = 0.1 millimetres per sheet. A reading error in the whole stack is also divided by one hundred when calculating the average. This method improves the measurement because the same reading difficulty becomes smaller compared with the total thickness. It does not guarantee that every sheet has exactly the same thickness. What we have determined is an average value. Measuring multiples is useful whenever one item is too small to measure well directly.

3:20For time intervals, choose an instrument suited to the event. A clock can measure a long interval by subtracting the starting time from the finishing time. A digital stopwatch or timer is more suitable for an event lasting seconds. Decide exactly what counts as the start and finish before taking a reading. For example, a lesson starting at nine fifteen and ending at ten gives an interval of forty-five minutes.

3:42For a short race, start and stop a stopwatch at the agreed events. Human reaction time introduces uncertainty because button presses may happen slightly before or after those events. A digital display does not remove this human limitation. A pendulum provides another example of measuring multiples. One complete oscillation is a movement from a starting position and back to that position, moving in the same direction. From one extreme, it swings to the opposite extreme and returns.

4:09The period is the time taken for one complete oscillation, not one outward swing. Start timing at a clearly recognised position, such as one extreme, and begin the count at zero. Count one when the pendulum returns after its first complete oscillation. Continue until twenty complete oscillations have occurred, then stop at the same position. Dividing the total time by twenty gives the average period. Suppose twenty oscillations take thirty-two seconds.

4:37Period equals total time divided by the number of complete oscillations. 32 ÷ 20 = 1.6 seconds. Timing many oscillations reduces the effect of reaction-time uncertainty compared with timing just one, and repeating the measurement helps check consistency.

Scalars and vectors

4:54We now move to Supplement content for the Extended course. A scalar quantity has magnitude only, while a vector quantity has magnitude and direction. Magnitude means size, expressed using a number and an appropriate unit. Direction matters for a vector because changing its direction changes the quantity, even when its magnitude stays the same. The scalars you need to know are distance, speed, time, mass, energy and temperature. Each describes a quantity without requiring a direction.

5:22For example, ten seconds fully specifies a time interval. A temperature below zero is still a scalar, because a negative value is not a direction in space. The required vectors include force, weight, velocity and acceleration. Momentum, electric field strength and gravitational field strength are also vectors. You will meet their detailed meanings in later videos. For now, recognise that each has magnitude and direction. Weight belongs in the vector list because it is a force, whereas mass is a scalar.

5:52Speed and velocity show why the distinction matters. Five metres per second describes speed, but five metres per second east specifies velocity. Two objects travelling equally fast in opposite directions have equal speeds but different velocities.

Resultant forces at right angles

6:07When representing a vector with an arrow, its length shows magnitude and its arrowhead shows direction. The resultant is a single vector representing the combined effect of two or more vectors. Here, we consider only two forces or two velocities at right angles. Imagine a force of three newtons east and another of four newtons north. Neither force points in the resultant direction, because their combined effect lies between east and north.

6:31Draw the first arrow, then place the tail of the second at the head of the first. Keep both arrows' magnitudes and directions unchanged. The resultant runs from the starting tail to the final head. Because the original vectors are perpendicular, these three arrows form a right-angled triangle. The resultant is the hypotenuse, which is the side opposite the right angle. Pythagoras states that its square equals the sum of the squares of the other two sides.

6:56Square three to get nine, and square four to get sixteen. Add these to get twenty-five, then take the square root to obtain a resultant magnitude of five newtons. Adding three and four directly would give seven newtons, but that ignores direction. Ordinary addition gives the resultant magnitude when both forces point in the same direction. These forces point at right angles, so the diagonal represents their combined effect. The resultant's direction lies between the two original directions.

7:24You can also determine the resultant graphically using a scale drawing. Choose one centimetre to represent one newton, then draw three centimetres east followed by four centimetres north. Join the starting point to the final point and measure that diagonal. Its length is five centimetres, representing five newtons, and a protractor gives approximately fifty-three degrees north of east. The reference direction must be stated when giving an angle. Fifty-three degrees north of east means measuring from east towards north.

7:54Measuring from north instead gives approximately thirty-seven degrees east of north. Both describe the same arrow, but an angle without its reference direction is incomplete.

Velocity resultants and recap

8:04The same method works for perpendicular velocities. Suppose a boat's velocity relative to the water is six metres per second east, while the water flows eight metres per second north. Square six to obtain thirty-six, and square eight to obtain sixty-four. Adding gives one hundred, whose square root is ten, so the resultant speed is ten metres per second. To specify the resultant velocity fully, we also need its direction.

8:29A scale drawing using one centimetre for every two metres per second gives sides of three and four centimetres. Measure the diagonal's angle from east to obtain approximately fifty-three degrees north of east. The resultant velocity is ten metres per second in that direction. In an examination, you might be asked to describe how to determine a pendulum's period accurately. A clear answer states that you time twenty complete oscillations using a stopwatch.

8:53Start and stop at the same position, counting complete returns correctly. Divide the measured time by twenty, and repeat the measurement to calculate a mean period. Other questions may ask you to read a scale, calculate a displaced volume, classify quantities or determine a resultant. Show the subtraction or division needed, and include the correct unit. For a vector answer, give magnitude and direction where requested. Keep ruler readings, complete oscillations and vector directions clear, because each represents a different part of the measurement.

9:24The central idea is to match the method to the quantity being measured. Read scales without parallax, and measure multiples when individual distances or intervals are too small. Scalars have magnitude only, while vectors have magnitude and direction. For perpendicular forces or velocities, a right-angled vector triangle gives the resultant by calculation or scale drawing.