Question 1

In microeconomics, what is the arc elasticity of demand when quantity demanded falls from 1000 to 950 units as price rises from Rs 240 to Rs 280?

Accepted Answer

Introduction / Context: Arc elasticity of demand is a widely used concept in microeconomics and business studies to measure how responsive quantity demanded is to a change in price over a finite range. Instead of using point elasticity at a single price quantity combination, arc elasticity uses average price and average quantity across two points on a demand curve. This makes it very useful for analysing real data from markets where prices and quantities change between two levels rather than by tiny amounts. Given Data / Assumptions: • Initial quantity demanded Q1 = 1000 units. • Final quantity demanded Q2 = 950 units. • Initial price P1 = Rs 240. • Final price P2 = Rs 280. • Arc elasticity of demand is defined using average price and average quantity over the two points. Concept / Approach: The formula for arc price elasticity of demand E on a straight segment of the demand curve is E = (ΔQ / ΔP) * (P_avg / Q_avg). Here ΔQ is Q2 minus Q1, ΔP is P2 minus P1, P_avg is the average of P1 and P2, and Q_avg is the average of Q1 and Q2. For a normal downward sloping demand curve, when price increases, quantity demanded falls, so ΔQ and ΔP have opposite signs and the elasticity is negative. In exam answers, it is common to retain the negative sign to reflect the inverse relationship. Step-by-Step Solution: Step 1: Compute the change in quantity demanded, ΔQ = Q2 − Q1 = 950 − 1000 = −50 units. Step 2: Compute the change in price, ΔP = P2 − P1 = 280 − 240 = 40 rupees. Step 3: Compute the average quantity, Q_avg = (Q1 + Q2) / 2 = (1000 + 950) / 2 = 1950 / 2 = 975 units. Step 4: Compute the average price, P_avg = (P1 + P2) / 2 = (240 + 280) / 2 = 520 / 2 = 260 rupees. Step 5: Substitute into the arc elasticity formula, E = (ΔQ / ΔP) * (P_avg / Q_avg) = (−50 / 40) * (260 / 975). Step 6: Simplify −50 / 40 to −5 / 4 and simplify 260 / 975 by dividing numerator and denominator by 65, giving 4 / 15. Step 7: Multiply the simplified fractions, E = (−5 / 4) * (4 / 15) = −5 / 15 = −1 / 3. Step 8: Convert −1 / 3 to decimal form, which is approximately −0.33. Verification / Alternative check: As a quick reasonableness check, notice that price increased by about 16 to 17 percent from 240 to 280, while quantity fell by 50 units out of 975 on average, which is a fall of a little over 5 percent. A 5 percent fall in quantity for about a 16 percent rise in price suggests an elasticity magnitude a little less than 0.4, which is close to one third. The sign must be negative because the quantity moved in the opposite direction to price. This sanity check supports the detailed calculation of about −0.33 as the correct arc elasticity. Why Other Options Are Wrong: 0.33: This ignores the negative sign and therefore hides the inverse relationship between price and demand, which is not standard in theory questions. 0.3: This is a rough number but not the correct value from the exact formula and also ignores the sign. −0.3: The sign is correct but the magnitude is slightly off compared with the more precise calculation of one third. 0: This would mean quantity demanded did not react at all to the price change, which is clearly not true because quantity did fall from 1000 to 950. Common Pitfalls: A common mistake is to use only initial price and initial quantity rather than average values, which gives point elasticity rather than arc elasticity. Another frequent error is to forget the negative sign when price increases and quantity falls. Some students also mix up percentage changes and absolute changes, or they round intermediate steps too early, leading to inaccurate final values. To avoid these issues, always write the formula clearly, compute ΔQ and ΔP with attention to sign, use average price and average quantity, and simplify the fractions step by step before converting to decimal form. Final Answer: The arc elasticity of demand is approximately −0.33.

Question 2

Why does a fountain pen often leak inside an aeroplane flying at high altitude?

Accepted Answer

Introduction / Context: Many everyday observations can be explained using basic physics of pressure and fluids. One classic example is that a fountain pen, which works normally on the ground, may start leaking ink when carried in an aeroplane flying at high altitude. Understanding why this happens connects concepts of atmospheric pressure, pressure inside a container and the behaviour of liquids and trapped air when the surrounding pressure changes. Given Data / Assumptions: • The pen is a fountain type or ink pen with ink and trapped air inside its reservoir. • The pen was filled and sealed at normal atmospheric pressure on the ground. • The aeroplane is flying at high altitude where cabin pressure is lower than sea level pressure, even though it is partially regulated. • Temperature changes are not large enough to be the main effect in this basic explanation. Concept / Approach: The key idea is the relation between pressure inside the pen and pressure outside in the cabin. When the pen is filled at ground level, the air above the ink is at roughly one atmosphere of pressure. As the aeroplane climbs, even in a pressurised cabin, the air pressure outside the pen decreases. The air trapped inside the reservoir now has a pressure greater than the surrounding air. Gases expand when external pressure drops. The expanding air inside the pen pushes on the ink, forcing it out through the nib and feed. This extra push overcomes the normal balance that keeps the ink in place and causes leakage. Step-by-Step Solution: Step 1: Consider the pen when it is filled and closed at ground level where outside atmospheric pressure is high. Step 2: Some air remains trapped above the ink and this enclosed air has the same pressure as the outside air at that moment. Step 3: When the aeroplane climbs to cruising altitude, the cabin pressure is reduced compared with ground pressure. Step 4: The trapped air inside the pen now has higher pressure than the air outside in the cabin. Step 5: This higher internal pressure makes the trapped air expand, which in turn pushes the ink out through the nib, leading to leakage. Verification / Alternative check: A similar effect is observed in other closed containers taken to lower external pressure, such as loosely sealed plastic bottles that bulge out slightly when taken to high mountains. In all these cases, the pressure inside was set at a higher external pressure at the time of sealing. As external pressure falls, the imbalance leads to expansion or leakage. For pens, manufacturers often advise travellers either to carry them empty or to keep them in a nib up position during flights, which is a practical confirmation that pressure differences are the main cause. Why Other Options Are Wrong: Reduced viscosity of the ink: A moderate change in temperature does not reduce viscosity enough to suddenly cause leakage in this way, and viscosity changes do not explain directional flow outwards. Increased viscosity of the ink: Higher viscosity would actually make the ink flow more slowly, so it would help to prevent leakage, not cause it. Higher atmospheric pressure outside the pen: In an aeroplane at altitude, cabin pressure is lower than ground level, so this statement is opposite to reality. Very high temperature inside the cabin: Cabin temperature is controlled for passenger comfort and does not become high enough to directly cause such leaking behaviour on its own. Common Pitfalls: Learners sometimes blame temperature alone or think the ink boils, but modern aircraft cabins are temperature controlled and the main physical change is in pressure. Another confusion comes from mixing up cabin pressure with conditions outside the aircraft; although cabins are pressurised, they still operate at a lower pressure than sea level. The safest way to reason through such problems is to compare pressure inside a container with pressure outside and then apply the simple rule that gas expands when external pressure decreases, pushing on any liquid in contact with it. Final Answer: A fountain pen leaks in an aeroplane mainly because the atmospheric pressure outside the pen is lower at altitude, so the higher pressure air trapped inside expands and pushes the ink out.

Question 3

Which meteorological instrument is used to measure wind speed or wind velocity?

Accepted Answer

Introduction / Context: Weather observation and forecasting rely on several specialised instruments that measure different properties of the atmosphere. One important quantity for aviation, shipping, agriculture and sports is wind speed or wind velocity. The instrument name that appears very often in competitive exams is the one designed specifically to measure how fast the wind is blowing at a particular location. Given Data / Assumptions: • The question asks about an instrument for measuring wind speed or wind velocity. • It is a conceptual general knowledge question with no numerical data. • The names given in the options refer to different scientific instruments used in various fields. Concept / Approach: In meteorology, the standard instrument used to measure wind speed is called an anemometer. The most common type is the cup anemometer, which has three or four hemispherical cups mounted on horizontal arms that rotate around a vertical shaft. As the wind blows, it turns the cups, and the rotation speed is calibrated to give the wind speed. Other types include propeller anemometers and ultrasonic anemometers, but all of them fall under the general term anemometer. By contrast, the other instrument names in the options are associated with different physical quantities. Step-by-Step Solution: Step 1: Identify the physical quantity mentioned in the question, which is wind speed or wind velocity. Step 2: Recall the standard meteorological instrument used for this purpose, which is called an anemometer. Step 3: Compare this with the list of options provided in the question. Step 4: Select the option that matches the instrument name anemometer. Step 5: Confirm that none of the other options are used for wind speed measurement. Verification / Alternative check: You can cross check by recalling that weather stations usually use a combination of a wind vane and an anemometer. The wind vane shows wind direction, while the anemometer measures speed. Meteorological diagrams and textbook illustrations of weather instruments consistently label the rotating cup device as an anemometer. This consistency across different sources confirms that the association of anemometer with wind velocity is correct. Why Other Options Are Wrong: Coulombmeter: This refers to an instrument for measuring electric charge in coulombs and is used in electricity and electronics experiments, not in meteorology for wind speed. Cyanometer: This is a historical instrument used to measure the blueness of the sky, related to atmospheric clarity, not wind velocity. Chronometer: A chronometer is a highly accurate timekeeping device, often used for navigation at sea. It measures time, not wind speed. Barometer: A barometer is used to measure atmospheric pressure and is crucial for weather forecasting, but it does not give wind speed readings. Common Pitfalls: Students sometimes confuse barometer and anemometer because both are meteorological instruments and they may have learned them together. Another common mistake is to answer wind vane instead of anemometer in questions that mention wind, but a wind vane measures direction, not speed. To avoid these errors, it helps to link the root ana for wind in anemometer with its function and remember that barometers are always connected with pressure and weather maps of high and low pressure. Final Answer: The instrument used to measure wind speed or wind velocity is the anemometer.

Question 4

In rotational mechanics, what is the perpendicular distance between the line of action of a force and the axis of rotation called?

Accepted Answer

Introduction / Context: Rotational motion and turning effects of forces are central topics in mechanics. When a force is applied at some distance from an axis of rotation, it produces a turning effect known as a moment or torque. The size of this turning effect depends not only on the magnitude of the force but also on how far the line of action of the force is from the axis. The special name for this perpendicular distance is an important term that often appears in theory and numerical questions. Given Data / Assumptions: • We have a rigid body that can rotate about an axis. • A force acts on the body at some point, and its line of action does not pass through the axis. • The perpendicular distance from the axis to the line of action of the force is considered. • We seek the correct technical term used in physics for that perpendicular distance. Concept / Approach: The turning effect or moment of a force about an axis is given by the expression Moment = Force * perpendicular distance from the axis. This perpendicular distance from the axis of rotation to the line of action of the force is known as the moment arm or lever arm. It is crucial that the distance measured is the shortest distance, hence perpendicular. This is different from the straight line distance from the axis to the point of application when the force is not perpendicular. Other terms like moment of inertia and torque refer to different concepts and should not be confused with the distance itself. Step-by-Step Solution: Step 1: Recall the formula for the moment or torque of a force about an axis, which is torque = F * r_perp. Step 2: Identify r_perp as the perpendicular distance from the axis of rotation to the line of action of the force. Step 3: Recall the terminology used for this distance, which is called the moment arm or lever arm. Step 4: Examine the given options and search for the term that matches this definition. Step 5: Recognise that moment arm is the correct name for this perpendicular distance. Verification / Alternative check: You can verify this understanding from practical examples such as using a spanner to turn a nut. The longer the handle, the larger the moment arm for the same applied force and the easier it becomes to turn the nut. This everyday experience is often explained in textbooks with the phrase longer lever arm or larger moment arm. The same idea is applied in levers, doors and see saws. In all these cases, it is the perpendicular distance from the pivot or hinge that determines the turning effect. Why Other Options Are Wrong: Moment of inertia: This is a measure of how much a body resists angular acceleration and depends on mass distribution, not a simple distance by itself. Altitude: This word is used mainly in geometry for the height of a triangle or in geography for height above sea level, not for torque calculations. Base: Base is also a geometric term for one side of a figure and does not denote the perpendicular distance from an axis to a force line in mechanics. Torque: Torque or moment is the product of the force and the moment arm. It is the turning effect, not the distance that enters the formula. Common Pitfalls: Students sometimes confuse moment arm with moment of inertia because both appear in rotational motion formulas. Another mistake is to think the distance used in torque calculations is just the straight line distance from the axis to the point of application, even when the line of action of the force is not perpendicular. The correct procedure is always to take the perpendicular distance from the axis to the force line. Drawing a clear diagram and marking the right angle distance helps to avoid these errors. Final Answer: The perpendicular distance from the axis of rotation to the line of action of a force is called the moment arm or lever arm.

Question 5

For which of the following physical quantities is the SI unit not the pascal?

Accepted Answer

Introduction / Context: In mechanics and material science, many different physical quantities are measured with SI units that may look similar. Pressure, stress and various elastic moduli are closely related, and all of them involve force over area. The pascal is the SI unit that appears repeatedly in this group. However, not every mechanical quantity uses the pascal, and competitive exams often ask you to pick the one that does not fit. Given Data / Assumptions: • The pascal is defined as one newton per square metre, written as N/m^2. • The question lists several quantities: moment of inertia, pressure, stress, Young modulus and bulk modulus. • We must identify which quantity does not use the pascal as its SI unit. Concept / Approach: Pressure is force per unit area, so its SI unit is N/m^2, which is the pascal. Stress is also defined as internal force per unit area within a material, so it has the same SI unit, the pascal. Young modulus and bulk modulus describe ratios of stress to strain. Since strain is dimensionless, the units of these moduli are identical to the units of stress, again pascal. Moment of inertia is completely different. It involves mass multiplied by the square of distance and is measured in kilogram metre squared. Step-by-Step Solution: Step 1: Recall that pressure = force / area, so its unit is newton per square metre, that is the pascal. Step 2: Recall that stress is internal force / area, so its unit is also newton per square metre, the pascal. Step 3: Recall that Young modulus = stress / strain, and since strain has no units, Young modulus has the same unit as stress, the pascal. Step 4: Bulk modulus is also a ratio of stress to fractional volume change, so again its unit is the pascal. Step 5: Moment of inertia depends on mass and length squared and has SI unit kilogram metre squared, not pascal, so this is the correct choice. Verification / Alternative check: You can verify quickly by dimensional analysis. Force has dimensions of mass times acceleration, so M L T^-2. Area has dimensions L^2. Pressure therefore has dimensions M L^-1 T^-2. The same dimensions apply to stress and all elastic moduli derived from it. In contrast, moment of inertia is defined as the sum or integral of m * r^2, so it has dimensions M L^2. Because these dimensional patterns are completely different, their SI units must also be different. This confirms that moment of inertia is the odd one out. Why Other Options Are Wrong: Pressure: By definition force per unit area, so its unit is N/m^2, which is exactly one pascal. Stress: Internal force per unit area inside a body, same unit N/m^2, so also measured in pascal. Young modulus: Ratio of stress to strain, strain has no units, so Young modulus has the same unit as stress, that is the pascal. Bulk modulus: Also a stress based modulus, defined as pressure change divided by fractional volume change, and therefore it has unit pascal. Common Pitfalls: A frequent error is to think that because a modulus sounds like a different type of quantity, it must have some special unit. In fact, whenever you divide a quantity with units by a dimensionless quantity, the units remain unchanged. Therefore all stress based moduli share the pascal as their SI unit. Another mistake is confusing moment of inertia with mass or weight. Remember that moment of inertia always involves a distance squared factor, which immediately distinguishes its unit as kilogram metre squared. Final Answer: Moment of inertia is not measured in pascal; its SI unit is kilogram metre squared, whereas pressure, stress and the elastic moduli all use the pascal.

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