# Questions for Measurement (JC) Set 1

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The mass M of the largest stone which can be moved by water of a flowing river is assumed to be
M = kvrpgs, where
k = dimensionless constant
v = velocity of water flow
p = density of water
g = acceleration due to gravity
and r, s are dimensionless exponents.

Find value of r.

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Comparing the base units of LHS and RHS of the equation:
Units of LHS = kg
Units of RHS = (ms-1)r(kg m-3)(ms-2)s
= m(r-3+s)s(-r-2s) kgr – 3 + s = 0 —– Eqn 1
-r -2s = 0 —– Eqn 2
Solving Eqn 1 and 2, r = 6

Which one of the following groups contains only vector quantities?

1. force, work, energy
2. displacement, momentum, acceleration
3. power, energy, time
4. momentum, mass, velocity
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Vector quantities: Force, displacement, momentum, velocity, acceleration
Scalar quantities: Work, energy, power, time, mass

The relationship between four physical quantities z, p, q and t, is given by

z2 = p + qt

, where t is the time in seconds.

If q has the units m s-1, then p must have the units

1. m s-1
2. m s-2
3. m s
4. m
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z2 = p + qt
units of qt → ms-1s = m

Hence, units of p is also m

a) (i) By Newton’s Second law of motion, the resultant force F acting on a constant mass m is given by F = ma, where a is the acceleration of the mass.

A student expresses the unit for mass as N s2m-1. Explain why its inappropriate to express mass in this unit.

ii) A student defines speed as “distance travelled per second”. Explain clearly why this statement is technically wrong. Hence, write down the correct definition for this speed.

b) In an experiment to measure the diameter of a uniform wire using a micrometer screw gauge, a student fails to notice that with the gauge fully closed, the reading is not zero.

i) State and explain clearly whether the omission introduces a random error or a systematic error into the readings of the diameter.

ii) State the effect(s), if any, the omission has on the precision and accuracy of the readings.

c) A spinning ball moving with a speed v in air of density ρ experiences a difference in pressure Δp on its sides. This causes its path of motion to e deflected.

i) By comparing the powers of the base units, suggest a possible relationship between the difference in pressure Δp and the quantities v and ρ.

ii) Suggest a reason why the relationship that you have found in (c)(i) may not be physically correct.

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a) i) Mass is a base quantity and has the base unit, kg. It is not a derived quantity and so it is inappropriate to express the unit for mass as N s2m-1. Furthermore, the defining equation for mass is not force per unit acceleration.
ii) Distance is a physical quantity while second is a unit. The physical quantity speed should be defined in terms of quantities, and not a mixture of a quantity and a unit. The correct definition for speed is the distance travelled per unit time.

b) i) The omission introduces a systematic error (zero error). This is because there is a constant error in one direction only, that is, it constantly causes the readings to be either larger or smaller than the true value.

b) ii) It causes the readings to be precise but not accurate.

c) i) Let Δp = kpxvy, where k,x and y are dimensionless constants.

units of Δp = units of kpxvy
(kg)(m s-2)(m-2) = (kg m-3)x(m s-1)y
kg m-1s-2 = kgxm-3x+ys-y

By comparing powers,
x = 1 and y = 2

ii) An equation that is dimensionally correct may be physically incorrect because there might be missing coefficients in the equation, or there might be terms which are missing.

Express the pascal in SI base units.

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The pascal is a unit for pressure.P = F/AIn SI base units,
Pa = (kg m s-2)/(m2)
= kg m-1s-2

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