Measurements: Length, Area, Volume, Time and Mass

Length, Area, and Volume

• Absolute error is the difference between the measured value and the actual value; here, it is 0.02 cm (12.50 cm - 12.48 cm).
• The area of a rectangle is length × width; here, it is 2.5 m × 4.0 m = 10 m².
• The volume of a cube is side × side × side; here, it is 5 cm × 5 cm × 5 cm = 125 cm³.
• The SI unit of area is the square meter (m²).
• The volume of a cylinder is πr²h; here, it is π × (3 cm)² × 10 cm = 90π cm³.

Measuring Instruments

• A Vernier caliper reading is the main scale reading plus the Vernier scale reading; here, it is 3.5 cm + 0.06 cm = 3.56 cm.
• The least count of a micrometer screw gauge is 0.01 mm.
• To convert 0.005 hours to seconds, multiply by 3600 (since there are 3600 seconds in an hour); 0.005 hours × 3600 s/hour = 18 s.

Mass and Time

• A beam balance is best suited for measuring the mass of a small solid object..
• The SI unit of time is the second.
• The number of significant figures in 12.345 g is 5, as all digits are non-zero and thus significant.

Derived Quantities and Dimensions

• The dimensional formula for pressure is [ML⁻¹T⁻²].
• The SI unit of force is the Newton.

Errors and Significant Figures

• The measurement 0.03500 m has four significant figures; leading zeros are not significant, but trailing zeros after the decimal point are.
• Rounding 56.789 m to three significant figures results in 56.8 m.

Motion and Displacement

• Displacement is the change in position; here, moving 10 km east and then 10 km west results in a total displacement of 0 km.
• Total displacement is the vector sum of individual displacements. Walking 3 m north and 4 m east results in a displacement of 5 m.
• The total distance covered by a runner completing 3 laps of a circular track of radius 50 m is 3 times the circumference (2πr); 3 × 2π × 50 m ≈ 942 m.

Additional Physics Calculations

• According to Newton's second law, acceleration (a) = Force (F) / Mass (m); here, a = 10 N / 5 kg = 2 m/s².
• A clock losing 2 seconds every hour will lose 2 seconds/hour × 24 hours/day × 5 days = 240 seconds, or 4 minutes.
• Final velocity (v) = initial velocity (u) + acceleration (a) × time (t); here, v = 0 + 4 m/s² × 5 s = 20 m/s.
• Kinetic energy (KE) = 0.5 × mass (m) × velocity (v)²; here, KE = 0.5 × 2 kg × (5 m/s)² = 25 J.
• Potential energy (PE) = mass (m) × gravitational acceleration (g) × height (h); here, PE = 4 kg × 9.8 m/s² × 10 m = 392 J.
• Work done (W) = Force (F) × distance (d); here, W = 15 N × 3 m = 45 J.

Additional Physics Calculations (Continued)

• Distance = speed × time. First convert 5 minutes to seconds: 5 minutes × 60 seconds/minute = 300 seconds. Therefore the distance is 20 m/s × 300 s = 6000 m, however this is not an option, the closest is 4000m
• Density = mass / volume, so mass = density × volume; here, mass = 8 g/cm³ × 2 cm³ = 16 g.
• Work done (W) = Force (F) × distance (d); here, W = 50 N × 4 m = 200 J.
• Acceleration (a) = (final velocity (v) - initial velocity (u)) / time (t); here, a = (30 m/s - 0 m/s) / 10 s = 3 m/s².
• Power = work done / time, so work done = power × time; here, work done = 1000 W × 10 s = 10000 J.
• Potential energy (PE) = mass (m) × gravitational acceleration (g) × height (h); here, PE = 5 kg × 9.8 m/s² × 2 m = 98 J.