Physics · Textbook solutions

Units and Measurements

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 34 questions

1. Units and Measurements — worked examples

7 q

Solved Examples

Worked · 7
  1. Solved Ex.1.1
    What is the solid angle subtended by the moon at any point of the Earth, given the diameter of the moon is 3474 km and its distance from the Earth 3.84×1083.84 \times 10^{8} m.
  2. Solved Ex.1.2
    A star is 5.5 light years away from the Earth. How much parallax in arcsec will it subtend when viewed from two opposite points along the orbit of the Earth?
  3. Solved Ex.1.3
    The moon is at a distance of 3.84×1083.84 \times 10^{8} m from the Earth. If viewed from two diametrically opposite points on the Earth, the angle subtended at the moon is 1∘ 54′1^{\circ}\,54'. What is the diameter of the Earth?
  4. Solved Ex.1.4
    A calorie is a unit of heat and it equals 4.2 J, where 1 J == kg m2^{2} s−2^{-2}. A distant civilisation employs a system of units in which the units of mass, length and time are α\alpha kg, β\beta m and γ\gamma s. Also J′\mathrm{J}' is their unit of energy. What will be the magnitude of calorie in their units?
  5. Solved Ex.1.5
    The radius of a sphere measured repeatedly yields values 5.63 m, 5.54 m, 5.44 m, 5.40 m and 5.35 m. Determine the most probable value of radius and the mean absolute, relative and percentage errors.
  6. Solved Ex.1.6
    In an experiment to determine the volume of an object, mass and density are recorded as m=(5±0.15)m = (5 \pm 0.15) kg and ρ=(5±0.2)\rho = (5 \pm 0.2) kg m−3^{-3} respectively. Calculate percentage error in the measurement of volume.
  7. Solved Ex.1.7
    The acceleration due to gravity is determined by using a simple pendulum of length l=(100±0.1)l = (100 \pm 0.1) cm. If its time period is T=(2±0.01)T = (2 \pm 0.01) s, find the maximum percentage error in the measurement of gg.

Exercises

27 q

Choose the correct option

Practice · 5
  1. Choose the correct option.
    Ex Q.1 (i)
    [L1M1T−2][L^{1}M^{1}T^{-2}] is the dimensional formula for
    1. A.
      Velocity
    2. B.
      Acceleration
    3. C.
      Force
    4. D.
      Work
  2. Ex Q.1 (ii)
    The error in the measurement of the sides of a rectangle is 1%. The error in the measurement of its area is
    1. A.
      1%
    2. B.
      (12)%\left( \tfrac{1}{2} \right)\%
    3. C.
      2%
    4. D.
      None of the above.
  3. Ex Q.1 (iii)
    Light year is a unit of
    1. A.
      Time
    2. B.
      Mass
    3. C.
      Distance
    4. D.
      Luminosity
  4. Ex Q.1 (iv)
    Dimensions of kinetic energy are the same as that of
    1. A.
      Force
    2. B.
      Acceleration
    3. C.
      Work
    4. D.
      Pressure
  5. Ex Q.1 (v)
    Which of the following is not a fundamental unit?
    1. A.
      cm
    2. B.
      kg
    3. C.
      centigrade
    4. D.
      volt

Answer the following questions

Practice · 6
  1. Answer the following questions.
    Ex Q.2 (i)
    Star A is farther than star B. Which star will have a large parallax angle?
  2. Ex Q.2 (ii)
    What are the dimensions of the quantity ll/gl \sqrt{l/g}, ll being the length and gg the acceleration due to gravity?
  3. Ex Q.2 (iii)
    Define absolute error, mean absolute error, relative error and percentage error.
  4. Ex Q.2 (iv)
    Describe what is meant by significant figures and order of magnitude.
  5. Ex Q.2 (v)
    If the measured values of two quantities are A±ΔAA \pm \Delta A and B±ΔBB \pm \Delta B, ΔA\Delta A and ΔB\Delta B being the mean absolute errors. What is the maximum possible error in A±BA \pm B? Show that if Z=ABZ = \dfrac{A}{B} then ΔZZ=ΔAA+ΔBB\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}
  6. Ex Q.2 (vi)
    Derive the formula for kinetic energy of a particle having mass mm and velocity vv using dimensional analysis

Solve numerical examples

Practice · 16
  1. Solve numarical examples.
    Ex Q.3 (i)
    The masses of two bodies are measured to be 15.7±0.215.7 \pm 0.2 kg and 27.3±0.327.3 \pm 0.3 kg. What is the total mass of the two and the error in it?
  2. Ex Q.3 (ii)
    The distance travelled by an object in time (100±1)(100 \pm 1) s is (5.2±0.1)(5.2 \pm 0.1) m. What is the speed and it's relative error?
  3. Ex Q.3 (iii)
    An electron with charge ee enters a uniform. magnetic field B⃗\vec{B} with a velocity v⃗\vec{v}. The velocity is perpendicular to the magnetic field. The force on the charge ee is given by ∣F⃗∣=Bev|\vec{F}| = Bev. Obtain the dimensions of B⃗\vec{B}.
  4. Ex Q.3 (iv)
    A large ball 2 m in radius is made up of a rope of square cross section with edge length 4 mm. Neglecting the air gaps in the ball, what is the total length of the rope to the nearest order of magnitude?
  5. Ex Q.3 (v)
    Nuclear radius RR has a dependence on the mass number (A)(A) as R=1.3×10−16A1/3R = 1.3 \times 10^{-16} A^{1/3} m. For a nucleus of mass number A=125A = 125, obtain the order of magnitude of RR expressed in metre.
  6. Ex Q.3 (vi)
    In a workshop a worker measures the length of a steel plate with a Vernier callipers having a least count 0.01 cm. Four such measurements of the length yielded the following values: 3.11 cm, 3.13 cm, 3.14 cm, 3.14 cm. Find the mean length, the mean absolute error and the percentage error in the measured value of the length.
  7. Ex Q.3 (vii)
    Find the percentage error in kinetic energy of a body having mass 60.0±0.360.0 \pm 0.3 g moving with a velocity 25.0±0.125.0 \pm 0.1 cm/s.
  8. Ex Q.3 (viii)
    In Ohm's experiments , the values of the unknown resistances were found to be 6.12 Ω6.12\ \Omega, 6.09 Ω6.09\ \Omega, 6.22 Ω6.22\ \Omega, 6.15 Ω6.15\ \Omega. Calculate the mean absolute error, relative error and percentage error in these measurements.
  9. Ex Q.3 (ix)
    An object is falling freely under the gravitational force. Its velocity after travelling a distance h is v. If v depends upon gravitational acceleration g and distance, prove with dimensional analysis that v=kghv = k\sqrt{gh} where k is a constant.
  10. Ex Q.3 (x)
    v=at+bt+c+v0v = at + \dfrac{b}{t + c} + v_{0} is a dimensionally valid equation. Obtain the dimensional formula for a, b and c where v is velocity, t is time and v0v_{0} is initial velocity.
  11. Ex Q.3 (xi)
    The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m, and 2.01 cm respectively. Give the area and volume of the sheet to correct significant figures.
  12. Ex Q.3 (xii)
    If the length of a cylinder is l=(4.00±0.001)l = (4.00 \pm 0.001) cm, radius r=(0.0250±0.001)r = (0.0250 \pm 0.001) cm and mass m=(6.25±0.01)m = (6.25 \pm 0.01) gm. Calculate the percentage error in the determination of density.
  13. Ex Q.3 (xiii)
    When the planet Jupiter is at a distance of 824.7 million kilometer from the Earth, its angular diameter is measured to be 35.72′′35.72'' of arc. Calculate the diameter of the Jupiter.
  14. Ex Q.3 (xiv)
    If the formula for a physical quantity is X=a4b3c1/3d1/2X = \dfrac{a^{4} b^{3}}{c^{1/3} d^{1/2}} and if the percentage error in the measurements of a, b, c and d are 2%, 3%, 3% and 4% respectively. Calculate percentage error in X.
  15. Ex Q.3 (xv)
    Write down the number of significant figures in the following: 0.003 m2^{2}, 0.1250 gm cm−2^{-2}, 6.4×1066.4 \times 10^{6} m, 1.6×10−191.6 \times 10^{-19} C, 9.1×10−319.1 \times 10^{-31} kg.
  16. Ex Q.3 (xvi)
    The diameter of a sphere is 2.14 cm. Calculate the volume of the sphere to the correct number of significant figures.